%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2025}
\articlenumber{5}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}

\usepackage{lipsum}
\usepackage[dvipsnames]{xcolor}
\usepackage{graphicx,enumitem, booktabs, array, verbatim, longtable} % Required for inserting images
\usepackage{tikz,tikz-cd}
\usepackage{csvsimple}
\usepackage{svg}
\usepackage{mathrsfs}
%
%\newcommand{\emanuele}[1]{{\color {Red} #1}}
%
%\usepackage[normalem]{ulem}
%
%%\usepackage{mathrsfs}
%%\usepackage{nameref}
\usepackage{kbordermatrix}
%%\usepackage{varioref}
%%\usepackage[colorlinks=true,
%%linkcolor=teal,
%%citecolor=Periwinkle %Apricot
%%]{hyperref}
%
%%\usepackage{cleveref}
%%\usepackage{todonotes}
%
%%\usepackage[style=alphabetic,doi=false,isbn=false,url=false,eprint=true,
%%backend=biber,giveninits=true,,maxcitenames=3,maxbibnames=8,hyperref]{biblatex}
%%\addbibresource{references_lipschitz_polytopes.bib}
%
\newcommand{\comma}{, }
\DeclareMathOperator{\conv}{conv}
\newcommand{\groupnamea}{\scshape{id}}
\newcommand{\groupnameb}{C_2}
\newcommand{\groupnamec}{V_1}
\newcommand{\groupnamed}{V_2}
\newcommand{\groupnamee}{D_8}
\newcommand{\groupnamef}{D_{10}}
\newcommand{\groupnameg}{D_{12}}

\usepackage{subcaption}
%\captionsetup[subfloat]{labelfont=normalfont}

\newcommand{\Q}{\mathbb{Q}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\W}{\mathcal{W}}
\newcommand{\FF}{\mathscr{F}}
\newcommand{\KK}{\mathscr{K}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\KRW}[1]{KRW(#1)}

\newcommand{\GPF}[1]{\FF_{#1}} % GP-fan
\newcommand{\GPFclosed}[1]{\overline{\FF}_{#1}} % GP-fan-closed
\newcommand{\metric}{\rho}
\newcommand{\CA}[1]{\W_{#1}}
\newcommand{\GPFS}[1]{\mathring{\FF}_{#1}} % GP-fan on strict
\newcommand{\mcone}[1]{\mathbf M_{#1}}
\newcommand{\mconeclosure}[1]{\overline{\mathbf M}_{#1}}
\newcommand{\kfan}[1]{\KK_{#1}} % fan of circular metrics / kalmanson metrics
\newcommand{\kfanclosed}[1]{\overline{\KK}_{#1}} % fan of kalmanson pseudometrics.
\newcommand{\splitpm}[2]{\delta_{#1,#2}}
\newcommand{\elsplitsclosed}[1]{\overline{\mathscr E}_{#1}}
\newcommand{\elsplits}[1]{\mathscr E_{#1}}

\newcommand{\tk}[2]{t^{(#1,#2)}}

%\numberwithin{equation}{section}

%\theoremstyle{plain}
\newtheorem{lemma}[equation]{Lemma}
\newtheorem{theorem}[equation]{Theorem}
\newtheorem*{Theorem}{Theorem}
\newtheorem{conjecture}[equation]{Conjecture}
\newtheorem{condition}[equation]{Condition}
\newtheorem{corollary}[equation]{Corollary}
\newtheorem{proposition}[equation]{Proposition}
\theoremstyle{definition}
\newtheorem{definition}[equation]{Definition}
\newtheorem{remark}[equation]{Remark}
\newtheorem{remarks}[equation]{Remarks}
\newtheorem{example}[equation]{Example}
\newtheorem{problem}[equation]{Problem}
\newtheorem{question}[equation]{Question}
\newtheorem{notation}[equation]{Notation}

\makeatletter
\newcommand{\eqnum}{\refstepcounter{equation}\textup{\tagform@{\theequation}}}
\makeatother

%% define your title in the usual way
\title[The Wasserstein Arrangement]{Combinatorial invariants of finite metric spaces and the Wasserstein arrangement}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author[]{\texorpdfstring{Emanuele Delucchi\thanks{\href{mailto:emanuele.delucchi@supsi.ch}{emanuele.delucchi@supsi.ch}}\addressmark{1}, Lukas~K\"uhne\addressmark{2}\thanks{\href{lkuehne@math.uni-bielefeld.de}{lkuehne@math.uni-bielefeld.de}. Lukas Kühne is partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- SFB-TRR 358/1 2023 -- 491392403 and SPP 2458 -- 539866293$\partial$\;supp.}, \and Leonie Mühlherr\addressmark{2}\thanks{\href{lmuehlherr@math.uni-bielefeld.de}{lmuehlherr@math.uni-bielefeld.de}. Leonie Mühlherr is partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- SFB-TRR 358/1 2023 -- 491392403$\partial$\;supp.}}{Emanuele Delucchi, Lukas~K\"uhne, and Leonie Mühlherr}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1} IDSIA USI-SUPSI, University of applied arts and sciences, Lugano, Switzerland \\ \addressmark{2}Universit\"at Bielefeld, Fakult\"at f\"ur Mathematik, Bielefeld, Germany}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{In 2010, Vershik proposed a new combinatorial invariant of metric spaces given by a class of polytopes that arise in the theory of optimal transport and are called ``Wasserstein polytopes'' or ``Kantorovich--Rubinstein polytopes'' in the literature. 
	Answering a question posed by Vershik, we describe the stratification of the metric cone induced by the combinatorial type of these polytopes through a hyperplane arrangement. Moreover, we study its relationships with the stratification by combinatorial type of the injective hull (i.e., the tight span) and, in particular, with certain types of metrics arising in phylogenetic analysis. We also compute enumerative invariants in the case of metrics on up to six points.}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{finite metric spaces, Lipschitz polytopes, hyperplane arrangements}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{references_muehlherr_102.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

	

	\section{Introduction}
	
	This article is centered around the study of {finite metric spaces} through the lens of polytopes and hyperplane arrangements. 
	Finite metric spaces arise in several applied contexts. Let us mention for instance  mathematical biology, where finite metrics  model genetic dissimilarities between different species \cite{Phylogenetics}. In this setting, a main research direction is to identify suitable classes of metric spaces and study the combinatorics and geometry of the associated subset of the metric cone, e.g., for geometric statistics. This field of research goes back to the study of phylogenetic trees \cite{Bandelt-Dress-Trees,BHV}, has grown to include more general phylogenetic networks \cite{HRS} and is presently very active (see e.g., \cite{Devadoss-petti,Francis_Steel,Fischer_Francis}).
	
	One of the combinatorial invariants of metric spaces that are widely used in the aforementioned applications as well as for theoretical considerations is their \emph{injective hull}, introduced by Isbell~\cite{Isbell} and rediscovered by Dress~\cite{Dress} under the name \emph{tight span}.
	The stratification of the metric cone by combinatorial type of tight spans is equivalent to the one determined by the subdivisions of the second hypersimplex studied by Sturmfels and Yu~\cite{SturmfelsYu} and recently by Casabella, Joswig, and Kastner~\cite{CJK24}.
	
	Motivated by the theory of optimal transport, Vershik described a   correspondence between finite metric spaces and a class of symmetric convex polytopes, the so-called \emph{Kantorovich--Rubinstein--Wasserstein (KRW) polytopes} or \emph{fundamental polytopes}~\cite{Vershik}. 
	\begin{definition}\label{def:KRW}
		The KRW polytope of an $n$-metric $\metric$ is a polytope in $\R^n$ defined as the convex hull:
		\[
		\KRW{\metric} = \conv\left\{\left. \frac{e_i-e_j}{\metric_{ij}} \right\vert 1\le i , j \le n\right\},
		\]
		where $e_i$ is the $i$-th standard basis vector of $\R^n$. From now on, we abbreviate $$p_{ij}:=\frac{e_i-e_j}{\metric_{ij}} \textrm{ for } i\neq j.$$
		The metric is called \emph{generic} if it is strict and the KRW polytope is simplicial.
	\end{definition}
	
	
	
	
	
	%\begin{example}
	If $\metric$ is the $n$-metric with $\metric_{ij}=1$ for all $i\neq j$, then $\KRW{\metric}$ is the type $A_n$ \emph{root polytope}.
%	%\end{example}
%	%\begin{example}
%	If $\metric$ is a graph-metric (i.e., there is a graph $G$ with vertex set $[n]$ such that $\metric_{ij}$ is the number of edges of a shortest path from $i$ to $j$ in $G$), then $\KRW{\metric}$ is the {\em symmetric edge polytope} of~$G$~\cite{dali-delucchi-michalek}, also known as the {\em adjacency polytope} of $G$ and, as such, appearing in different applied contexts   \cite{Chen,CDM}. If $\rho$ is a tree metric (i.e., it can be represented as the distances among a subset of the vertices of a metric tree), the number of faces of $\KRW{\metric}$ can be computed explicitly, and the dual polytope is a zonotope \cite{DelucchiHoessly}.
	%\end{example}
	
	
	
	
	The dual of a KRW polytope is the so-called \emph{Lipschitz polytope} \cite{Vershik}.
	This is a symmetric \emph{alcoved polytope}~\cite{LP1}; a class of polytopes that recently gained significant attention due to its connections to theoretical physics in the form of positive geometries and polypositroids~\cite{LP2}.
	
	\bigskip
	
	The problem of understanding the combinatorial structure (e.g., computing face numbers) of KRW and Lipschitz polytopes is open and significant for applications, see e.g., \cite{Becedasetal,venturelloetal}. 
	In this article, we focus on a related problem that goes back to a question by Vershik. 
	\begin{problem}[{\cite[``General Problem'', Section 1]{Vershik}}]
		Study and classify finite metric spaces according to the combinatorial properties of their KRW polytopes.
	\end{problem}
	
	The first progress on this question was achieved by Gordon and Petrov who gave a description of the face poset of KRW polytopes via linear inequalities on the values of the metric~\cite{GordonPetrov}, see~\Cref{sec:GP} for further details.
	This is the starting point for an in-depth study of KRW polytopes using the theory of hyperplane arrangements initiated in this paper.
	
	\subsection{The main results}
	We now summarize the main contributions of this article:
	\begin{enumerate}
		\item In \Cref{sec:WA} we introduce a \emph{Wasserstein arrangement} whose cells correspond to the combinatorial types of KRW polytopes. This arrangement builds upon the Gordon--Petrov description of the faces of KRW polytopes and could be of independent interest.
		%Moreover, via the above connection to alcoved polytopes, intersecting this arrangement with the metric cone naturally yields the type-fan of symmetric alcoved polytopes.
		
		\item Using the computer algebra system \texttt{OSCAR}~\cite{OSCAR-book} together with the Julia package \texttt{CountingChambers.jl}~\cite{BEK23} and the software \texttt{TOPCOM}~\cite{Rambau} we obtain an enumeration of the combinatorial types of KRW polytopes of generic metrics on $n=4,5,6$ points. This is displayed in Table \ref{sim_table}, where the number of combinatorial types of generic KRW polytopes -- i.e., the number of chambers of the Wasserstein arrangement --  is shown in the column titled ``Labeled''. The column ``Unlabeled'' shows the number of combinatorial types up to combinatorial isomorphisms %(where we are allowed to permute the vertices to obtain a combinatorial isomorphism)
		, which is the number of chambers of the Wasserstein arrangement modulo the natural action of the symmetric group $S_n$ via coordinate permutation.
		
		%This table already shows that the subdivision of the metric cone by combinatorial types of KRW polytopes is much finer than the subdivision by tight spans as for $n=6$ there are $194,160$ different tight spans which up to symmetry come in $339$ orbits~\cite{SturmfelsYu}.
		
		\begin{table}[th!] %h!
			\centering 
			
			\begin{tabular}{ccc}
				\toprule
				$n$  & Unlabeled & Labeled  \\ 
				\midrule 
				3  & 1 & 1  \\ 
				4 & 1 & 6  \\ 
				5 & 12 & 882 \\ 
				6 & 25,224 &  \\ 
				\bottomrule  
			\end{tabular} 
			\caption{Combinatorial types of KRW polytopes of generic $n$-metrics.} 
			\label{sim_table} 
		\end{table}
		
		\item We clarify the relation between KRW polytopes and tight spans in \Cref{sec:tight_spans} by providing examples of five-point metrics with isomorphic tight spans and combinatorially different KRW polytopes and vice versa. Hence, the two fan structures of the metric fan induced by the tight spans or the KRW polytopes are not refinements of each other.
		
		%\item We compare the fan structure induced by the Wasserstein arrangement on the metric cone with some types of metrics that have appeared prominently in the literature on phylogenetic analysis. %In particular, we show that the space of tree-like metrics and of circular-decomposable metrics are subfans of the Wasserstein fan (\Cref{prop:tl}, \Cref{thm:cd}), thus so is the simplicial fan of Circular Split Networks introduced by Devadoss and Petti \cite{Devadoss-petti} (and appearing also in work of Hacking, Keel and Tevelev \cite{HKT}), see \Cref{DP-HKT}. Moreover, the set of metrics satisfying the \emph{six-point condition} of Dress--Huber-Koolen--Moulton~\cite{SixPoints} is a subfan of the Wasserstein fan as well (\Cref{prop:6p}). However, the set of all totally split-decomposable metrics is not a subfan of the Wasserstein fan (see \Cref{example_decomp}), and neither is the set of all \emph{consistent, totally split-decomposable metrics} (\Cref{rem:consistent}). %We leave the following as open questions.
%		\begin{question}[Tree-based metrics] The notion of \emph{tree-based phylogenetic network} has gathered considerable attention in the literature on phylogenetic analysis, since its introduction by Francis and Steel in 2015  \cite{Francis_Steel,Francis_Semple_Steel}. The space of such tree-based phylogenetic networks has been the object of recent research by Fischer and Francis 
%			\cite{Fischer_Francis}.
%			In our context, it is therefore natural to ask whether the metric spaces arising from tree-based phylogenetic networks are a union of strata of the Wasserstein fan.
%		\end{question}
		
%		\begin{question}[Antipodal metrics]
%			A further type of metrics that has been studied by Dress, Huber and Moulton is the class of \emph{antipodal metrics} \cite{DHMantipodal}. Antipodality imposes strong constraints on the combinatorial structure of the tight span (e.g., a finite metric space is antipodal if and only if its tight span has a unique maximal cell, see \cite{HuKoMo1}). It is then natural to ask whether antipodal metric spaces form a subfan of the Wasserstein fan.
%		\end{question}
		
		\item We derive formulae for the number of faces of the KRW polytopes of non-generic strict metrics in \Cref{sec:strict_case}.
	\end{enumerate}
%	
%	\subsection{Roadmap of the article}
%	We introduce the main objects used throughout in~\Cref{sec:BG}.
%	The Wasserstein arrangement, capturing the combinatorial structures of KRW polytopes, is defined and discussed in~\Cref{sec:WA}.
%	In~\Cref{sec:triangulations} we explain the connection between KRW polytopes and a certain class of regular triangulations of type $A$  root polytopes.
%	Subsequently,~\Cref{sec:tight_spans} is devoted to the relation between KRW polytopes and tight spans.
%	We explore several noteworthy subfans of the Wasserstein arrangement stemming from phylogeny in~\Cref{sec:subfans}.
%	We present results concerning the $f$-vector of KRW polytopes associated with strict metrics in~\Cref{sec:strict_case}.
%	Lastly in~\Cref{sec:compuations}, we summarize our computational findings on KRW polytopes for metrics on up to $6$ points.
	
	\begin{remark}
	This extended abstract corresponds to an article that is published as preprint on the arXiv \cite{comb_inv}.
\end{remark}
	
%	\subsection*{Acknowledgments}
%	
%	We thank Nick Early and Benjamin Schr\"oter for fruitful discussions and Michael Joswig for helpful comments on an earlier version of this manuscript.
%	We are grateful to J\"org Rambau for providing us with relevant triangulations of the root polytope computed using \texttt{TOPCOM}.

%	
%	\section{The main characters}
%	\label{sec:BG}
%	
%	
	\subsection{The metric cone}
	
	We begin by defining finite metric spaces and setting up some terminology.
	
	
	\begin{definition}
		An $n$\emph{-metric} is a real symmetric $n\times n$ matrix~$\metric$ with entries $\metric_{ij}$ for $1\le i,j\le n$ satisfying 
		\begin{enumerate}
			\item $\metric_{ii}=0$ for all $1\le i \le n$,
			\item $\metric_{ij}>0$ for all $1\le i\neq j\le n$ and
			%		\item $\metric_{ij}=d_{ji}$ for all $1\le i < j \le n$ and
			\item $\metric_{ij}+\metric_{jk}\ge \metric_{ik}$ for all $1\le i,j,k\le n$.
		\end{enumerate}
	\medskip
	\noindent 
		The metric $\metric$ is called \emph{strict} if $\metric_{ij}+\metric_{jk}> \metric_{ik}$ for all $j\in \left[n\right]\setminus \{i,k\}$.
	\end{definition}
	
	Every $n$-metric $\metric$ is given as a symmetric $n\times n$ matrix with zero diagonal, thus it is determined by the $\binom{n}{2}$ values $\metric_{ij}$ with $i < j$.
	%This establishes a bijective correspondence between the set of all $n$-metrics and the following subset of  $\mathbb R^{\binom{n}{2}}$.
	
	
	\begin{definition} 
		Consider the vector space $\mathbb R^{\binom{n}{2}}$ with coordinates $x_{\{i,j\}}$ indexed by pairs of elements of $[n]$. The {\em metric cone} on $n$ elements is the subset $\mcone{n} \subseteq \mathbb R^{\binom{n}{2}}$ defined by 
		$$
		x_{\{i,j\}}>0, \quad x_{\{i,j\}} + x_{\{j,k\}} \geq x_{\{i,k\}},  \textrm{ for all pairwise distinct } i,j,k \in [n]. 
		$$
		We denote by $\mconeclosure{n}$ the (topological) closure of $\mcone{n}$. 
	\end{definition}
	
	
	\begin{remark}
		The space $\mconeclosure{n}$ is a polyhedral cone. It contains points that do not correspond to metrics but rather to pseudometrics (where $\metric_{ij}=0$ can be allowed). The set of all strict metrics on $n$ elements is the interior of $\mcone{n}$. 
	\end{remark}
	
	
	
%	\begin{example} \label{defsplits}
%		Let $A\uplus B$ be a nontrivial bipartition of $[n]$. The pseudometric $\splitpm{A}{B}$ with
%		$$
%		(\splitpm{A}{B})_{i,j}:= \min \{\vert A\cap\{i,j\}\vert,\vert B\cap\{i,j\}\vert \}
%		$$
%		%\emanuele{Das einzige Fall wo wir Metrik-Werte nicht mit subscripts bexeichnen...}
%		has value $0$ when $i$ and $j$ are in the same part of the bipartition and has value $1$ otherwise. Any metric of this form is called an \emph{$n$-split}, the partition itself $\sigma = A\uplus B$ is called a \emph{split}.
%	\end{example}
	
	
	\subsection{Admissible graphs and strict metrics
	} \label{sec:GP}
	Let $\metric$ be an $n$-metric and $F$ any face of the polytope $KRW(\metric)$.
	We associate with the face $F$ a directed graph $G(F)$ on the vertex set $[n]$ which contains the edge $(i,j)$ if the point $p_{ij}$ lies on $F$. Following \cite{GordonPetrov}, the collection of graphs of the form $G(F)$ is called \emph{the combinatorial structure} of $\KRW{\metric}$.
	
	
	\begin{theorem}[{\cite[Theorem 3]{GordonPetrov}}]\label{thm:gordon_petrov}
		Let $\metric$ be an $n$-metric and $G = ([n], E)$ be a directed graph with the set of vertices $[n]$. The following are equivalent: 
		\begin{enumerate}
			\item There exists a facet $F$ of the polytope $\KRW{\metric}$ containing every vertex $p_{ij}$ for $(i,j)\in~E$.
			\item For any array of directed edges $(x_i,y_i), 1\leq i \leq k$ of $G$ with all $x_i$ pairwise distinct and all $y_i$ pairwise distinct: 
			\begin{equation}\label{eq:gp_inequality}
				\sum_{i = 1}^k \metric_{x_iy_i} \leq \sum_{i= 1} ^k \metric_{x_iy_{i+1}},
			\end{equation} where $y_{k +1} = y_1$. 
		\end{enumerate}
	\end{theorem}
	
	\begin{definition} 
		A directed graph $G$ on the vertex set $[n]$ is called \emph{admissible} for an $n$-metric $\metric$ if it satisfies one of the two equivalent conditions of \Cref{thm:gordon_petrov}. 
		
	\end{definition}
	
%	In general, the distinctness of the $x_i$ and $y_i$ does not mean that the two sets are disjoint. We have $x_i=y_j$ for some $i$ and $j$ if and only if the sequence $(x_j,y_j)$, $(x_i,y_i)$ is a directed path of length $2$ in $G$.
%	
%	However, for certain metrics it suffices to consider disjoint sets of start- and endvertices. 
	\begin{lemma}\label{lem:strict_admissible_condition}
		If $\metric$ be a strict metric, the condition in \Cref{thm:gordon_petrov}.(2) can only be satisfied if
		
		$x_1,\ldots,x_k$, $y_1,\ldots,y_k$ are pairwise distinct.
		
	\end{lemma} 
%	\begin{proof}
%		
%		Let $\metric$ be strict and consider an instance of \Cref{thm:gordon_petrov}.(2) with    $x_i=y_j$ for some $i$ and $j$.
%		Then \eqref{eq:gp_inequality} applied to $(x_j,y_j)$, $(x_i,y_i)=(y_j,y_i)$ gives
%		\[\metric_{xy}+\metric_{yz} \leq \metric_{yy}+\metric_{xz} = \metric_{xz}\] and thus, $\metric_{xy}+\metric_{yz} = \metric_{xz}$ which contradicts the assumption on $\metric$ being strict. 
%	\end{proof}
	
%	\begin{corollary}\label{strict_nopaths} If $\metric$ is strict, admissible graphs do not contain any directed path with more than $2$ vertices.
%	\end{corollary}
	
	We close this section with a characterization of generic metric spaces (in the sense of \Cref{def:KRW}).

	\begin{theorem}[{\cite[Theorem 5]{GordonPetrov}}]
		A strict $n$-metric $\metric$ is generic if and only if for arbitrary $2k$ distinct points $x_1,\dots,x_k, y_1,\dots,y_k$ in $\left[n\right]$ the minimum of the terms \[\sum_{i = 1}^k \metric_{x_i, y_{\pi(i)}}\] is attained by a unique permutation $\pi\in S_k$.
	\end{theorem}
	
	
	\section{The Wasserstein arrangement and the Wasserstein fan}
	\label{sec:WA}
	
	
	We now define an arrangement of hyperplanes whose faces encode the combinatorial structures of KRW polytopes.
	
	
	\begin{definition}[Wasserstein arrangement]
		Let $n$ be a positive integer. We define a hyperplane arrangement in $\R^{\binom{n}{2}}$ as follows. 
		Given $1<k$ and $k$-tuples $\mathbf{a}=(a_1,\ldots,a_k)$, $\mathbf{b}=(b_1,\ldots,b_k)$ with $a_i,b_i\in[n]$ for all $i=1,\ldots,k$, define a hyperplane 
		
		$$
		H_{\mathbf{a,b}} :=\left\{x\in \R^{\binom{n}{2}}\left\vert  \sum_{i=1} ^kx_{\{a_i,b_i\}} = \sum_{i=1}^k x_{\{a_i,b_{i+1}\}}\right.\right\}
		\quad \textrm{where }b_{k+1}=b_1.
		$$
		
		The ``positive side'' of $H_{\mathbf{a,b}}$ is %the halfspace
		$$
		H_{\mathbf{a,b}}^+ :=\left\{x\in \R^{\binom{n}{2}}\left\vert \sum_{i=1,\ldots, k} x_{\{a_i,b_i\}} \leq \sum_{i=1,\ldots,k} x_{\{a_i,b_{i+1}\}}\right.\right\}.
		$$
		The \emph{Wasserstein arrangement} is then the set of hyperplanes
		$$
		\CA{n}:=\left\{H_{\mathbf{a,b}}  \left\vert
		\begin{array}{l}
			1<k\leq n,\,\, \mathbf{a,b}\in [n]^k\\
			a_1,\ldots, a_k,b_1,\ldots, b_k \textrm{ mutually distinct}
		\end{array}
		\right.\right\}.
		$$
	\end{definition}
\noindent 
The hyperplanes $H_{\mathbf{a,b}}$ in $\CA{n}$ were already mentioned as ``exceptional planes'' in the proof of Theorem 7 in \cite{GordonPetrov}.
	\begin{remark}
		In the case $ k = 2$, the tuple $(a_1,a_2)$ forms the same hyperplane when combined with either of the tuples $(b_1, b_2)$ or $(b_2, b_1)$. In order for the notion of positive side to be well-defined, we establish the convention that we choose the tuple starting with $\min\{b_1,b_2\}$. 
	\end{remark}
	\begin{definition}[Wasserstein fan]\label{defWF} 
		Let $\GPF{n}$ denote the fan determined by the intersections of the Wasserstein arrangement $\CA{n}$ with the metric cone $\mcone{n}$. 
		We call $\GPF{n}$ the \emph{Wasserstein fan}.
		Moreover, write $\GPFclosed{n}$ for the (polyhedral) fan of pseudometrics, defined by the intersections of $\CA{n}$ with the cone $\mconeclosure{n}$.
	\end{definition}
	
	\begin{remark}\label{FanArr}
%		The metric cone $\mcone{n}$ intersects every face of the Wasserstein arrangement~$\CA{n}$.
%		Indeed, let $p$ be any point on some face of $\CA{n}$.
%		The point $p+k\mathbf{1}$ for some $k\ge 0$ and $\mathbf{1}$ the all-ones vector in $\R^{\binom{n}{2}}$ lies on exactly the same hyperplanes of the arrangement $\CA{n}$ as the point $p$ since the defining equations involve the same number of variables on both sides of the equation.
%		But for $k$ large enough, we observe that the point $p+k\mathbf{1}$ is in $\mcone{n}$.
%		
%		Therefore,  t
The arrangement $\CA{n}$ and the fan $\GPF{n}$ have the same number of $i$-cells for any~$i$.
		In particular, we can count the number of maximal cells of the fan $\GPF{n}$ by counting the number of chambers of the arrangement $\CA{n}$.
	\end{remark}
	
%	\begin{example}
%		The Wasserstein arrangement $\CA{4}$ in $\R^{\binom{4}{2}}$ consists of the three hyperplanes given by the equations $ x_{\{1,3\}} + x_{\{2,4\}}=x_{\{1,4\}} + x_{\{2,3\}}, x_{\{1,2\}} + x_{\{3,4\}}=x_{\{1,4\}} + x_{\{2,3\}}, x_{\{1,2\}} + x_{\{3,4\}}=x_{\{1,3\}} + x_{\{2,4\}}$. 
%		
%		This arrangement has a $4$-dimensional lineality space. 
%		%, see~\Cref{lem:lineality_space}. \Cref{fig:arrangement4} represents$\CA{4}$ modulo its lineality space.
%	\end{example}
	
	
	\begin{remark}\label{rem:cycle_arr}
		Each hyperplane $H_{\mathbf{a,b}}$ corresponds to a cycle $C_{H_{\mathbf{a,b}}}$ of the complete graph $K_n$ determined by the sequence of vertices $a_1,b_1,a_k,b_k,\ldots,a_2,b_2$. %(see \Cref{fig:cycle_example} for an example with  $k=3$).
		%The edges in the cycle have alternating orientations (all vertices have indegree either 0 or 2) and thus, the two sides of the hyperplane $H_{\mathbf{a,b}}$ determine which set of edges of the cycle is admissible (the even or the odd edges of the cycle). Similar to the halfspace notation, we refer to the set of edges $\{(a_i, b_i)_{1\leq i\leq k}\}$ by $C_{H_{\mathbf{a,b}}}^+$ and write $C_{H_{\mathbf{a,b}}}^-$ for the other set. 
		
		%The indexing by the pair of $k$-tuples will be necessary in order to specify a well-defined positive side. 
		
%		\begin{figure}[hbt]
%			\centering 
%			\includegraphics[width=0.25\textwidth]{figures/Zeichnung_cycle_example} 
%			\caption{A hyperplane $H_{\mathbf{a,b}}$ with $k=3$ interpreted as cycle.}
%			\label{fig:cycle_example}
%		\end{figure}
%		
	\end{remark}
	
	
	\begin{proposition}\label{prop:size_wn}
		The arrangement $\CA{n}$ consists of
		\[
		|\CA{n}|=\frac{1}{2}\sum_{k=2}^{\lfloor \frac{n}{2}\rfloor} \binom{n}{2k}(2k-1)!
		\]
		many hyperplanes.
	\end{proposition}
\noindent 
	The proof of this proposition uses the aforementioned cycle characterization of the hyperplanes. 
%	\begin{proof}
%		As discussed in \Cref{rem:cycle_arr}, each hyperplane in $\CA{n}$ is determined by an oriented cycle in in the complete graph $K_n$ of even length at least four.
%		Such cycles arise by choosing $2k$ vertices amongst the $n$ vertices and then choosing a cyclic permutation of length $2k$.
%		There are $(2k-1)!$ such permutations.
%		As reversing the orientation of the cycle yields the same hyperplane, we count every hyperplane exactly twice.
%	\end{proof}
	
	
	
	
%	\begin{remark} \label{rem:wassersteinarr}
%		
%		To study the combinatorial types of strict metrics, it suffices to consider the intersection of the interior of $\mcone{n}$ with $\CA{n}$, since by \Cref{strict_nopaths} admissible graphs for strict metrics cannot contain directed paths with more than one edge.
%	\end{remark}
	
%	\begin{definition}
%		Recall \Cref{defsplits}. The pseudometrics of the form $\splitpm{\{i\}}{[n]\setminus \{i\}}\in \R^{\binom{n}{2}}$ for $i=1,\ldots,n$ are called {\em elementary splits}. 
%		
%		Let $W$ denote the linear span of the elementary splits and let  $\elsplitsclosed{n}:=\mconeclosure{n}\cap W$ be the set of all pseudometrics that are linear combination of elementary splits. 
%		Accordingly, we write $\elsplits{n}:=\mcone{n}\cap W$ for the cone of metrics that are combinations of elementary splits.
%	\end{definition}
%	\begin{remark} \label{rem_dimW}
%		The set of elementary splits is ``compatible'' in the sense of \cite{BandeltDress} (e.g., because the metric obtained as the sum of all elementary splits is represented by a tree with $n$ leaves, each at distance $1$ from a unique inner vertex), and thus linearly independent by \cite[Corollary 4]{BandeltDress}. Therefore, $\dim(W)=n$ and  $\elsplitsclosed{n}$ is an $n$-dimensional simplicial cone given as the convex hull of the rays generated by the elementary splits. 
%	\end{remark}
%	
%	
%	\begin{lemma}\label{lem:lineality_space}
%		The lineality space of the Wasserstein arrangement $\CA{n}$ is the $n$-dimensional subspace $W$ spanned by the elementary splits (i.e., $W=\bigcap \CA{n}$).
%	\end{lemma}
%	\begin{proof}
%		Let $L:=\bigcap \CA{n}$.
%		We first prove that $W\subseteq L$.
%		If a metric $\metric$ is in a hyperplane $H_{\mathbf{a,b}}$ and we define \[\metric' := \metric +\lambda \cdot \splitpm{\{i\}}{[n]\setminus \{i\}}\in \R^{\binom{n}{2}}\] for $\lambda > 0$ and some $i$, then the $\lambda$ appears in  an equal number of summands on both sides of the defining equation of $H_{\mathbf{a,b}}$ and therefore, $\metric'\in H_{\mathbf{a,b}}$. Thus, $W$ is contained in $H_{\mathbf{a,b}}$ for all $\mathbf{a}$ and $\mathbf{b}$ and is thus contained in the lineality space of the arrangement $\CA{n}$.
%		
%		Now it is enough to prove that $\dim(W)\geq\dim(L)$. Since $\dim(W)=n$ (\Cref{rem_dimW}), it is enough to find ${\binom{n}{2}} - n$ hyperplanes in $\CA{n}$ whose normal vectors are linearly independent. 
%		
%		For this, identify the elements of the metric space $[n]$ with the vertices of a regular $n$-gon  
%		and consider the set $D$ of all pairs of distinct elements of $[n]$ that define a diagonal (i.e., not an edge) of the $n$-gon. Thus $\vert D\vert = \binom{n}{2}-n$. For $\{i,j\}\in D$ consider
%		$
%		H_{(i,j+1),(j,i+1)}
%		$ (we understand the sum modulo $n$, so that $n+1=1$), with normal vector
%		$v_{ij}= e_{\{i,j\}} + e_{\{i+1,j+1\}} - e_{\{i,i+1\}} - e_{\{j,j+1\}}$.
%		
%		We claim that the set $\{v_{ij} \mid \{i,j\} \in D\}$ is a linearly independent family. In order to do so, we arrange these vectors as columns of a matrix.
%		Consider the following total ordering on the set of all pairs of distinct elements of  $[n]$: 
%		
%		\begin{align*}
%			\{1,2\}\prec \{2,3\}\prec \dots\prec \{n,1\}\prec \{1,3\}\prec \{2,4\}\prec \dots \prec  \{n,2\}\prec \{1,3\}\prec\{2,4\}\prec\dots
%		\end{align*}
%		Under the bijection of unordered pairs with sides and diagonals of a regular $n$-gon, this corresponds to ordering these segments in increasing order of length and, for sides and diagonals of the same length $k$, in the cyclic order starting with the diagonal $\{1,k\}$.
%		
%		Let $A$ be the $\binom{n}{2} \times \vert D \vert$-matrix whose columns are the vectors $v_{ij}$ listed in increasing $\prec$-order of $\{i,j\}$, and where the $k$-th row corresponds to the coordinate $e_{\{i,j\}}$ with $\{i,j\}$ in $k$-th place with respect to the ordering $\prec$. Then $A$ has the following block-form:
%		$$
%		A =\left(\begin{array}{c|c|c|c}
%			W_1 & W_2 & W_3 & \cdots \\\hline
%			A_{1} & 0 & 0 & \cdots \\
%			0 & A_2 & 0 & \cdots \\
%			0 & 0 & A_{3} & \cdots \\
%			\vdots &  & & \ddots \\
%		\end{array}\right);\quad
%		\textrm{ with }
%		A_j = \begin{pmatrix}
%			1 & 0 & 0 & \cdots \\
%			1 & 1 & 0 & 0 & \cdots \\
%			0 & 1 & 1 & 0 & \cdots \\
%			0 & 0 & 1 & 1 & \ddots \\
%			\vdots &  & \ddots & \ddots & \ddots
%		\end{pmatrix},
%		$$
%		and the matrices $W_i$ have $n$ rows.
%		Since clearly each $A_i$ has full rank, so does $A$. 
%		This proves that the $v_{ij}$ are linearly independent and thus that
%		\[
%		\dim(L) \leq \dim\left(\bigcap_{\{i,j\}\in D} H_{\{i,j\},\{i+1,j+1\}}\right) = {\binom{n}{2}}- \vert D \vert =n.\qedhere
%		\]
%	\end{proof}
	\noindent
	The next proposition relates the combinatorial structure of KRW polytopes with the face structure of the Wasserstein fan.
	\begin{proposition}
		Let $n\ge 2$ be an integer and consider two $n$-metrics $\metric^{(1)},\metric^{(2)}$.
		The following two statements are equivalent:
		\begin{enumerate}
			\item The metrics $\metric^{(1)},\metric^{(2)}$ are in the open part of one cone $\sigma$ of the Wasserstein fan $\GPF{n}$.
			\item 
			The combinatorial structures of the polytopes $\KRW{\metric^{(1)}}$ and $\KRW{\metric^{(2)}}$ are the same, i.e., both polytopes have the same set of admissible directed graphs.
		\end{enumerate}
	\end{proposition}
%	\begin{proof}
%		Following \Cref{thm:gordon_petrov},
%		the sign pattern of a metric on an open cone in the fan $\GPF{n}$, i.e., describing whether the metric lies on, ``under'' or ``above'' a hyperplane, specifies the collection of admissible graphs of the metric.
%		As the latter is the combinatorial structure of the KRW polytope of the metric, the statement follows.
%	\end{proof}
	
	
	\noindent 
%	\section{Regular triangulations and the secondary fan}\label{sec:triangulations}
%	Following~\cite{DLRS} we start by recalling the definition of a regular subdivision of a set of points in $\R^n$, in particular of a convex polytope.
%	\begin{definition}\label{defRS}
%		Let $\cP=\{a_1,\dots,a_k\}$ be  set of points in $\R^n$.
%		A point $\omega\in \R^k$ defines a \emph{regular subdivision} $\cP(\omega)$ as follows:
%		A subset $\{{i_1},\dots,{i_r}\}$ is a facet of $\cP(\omega)$ if there exists a vector $c\in \R^n$ such that
%		\begin{equation}\label{eq:subdivision}
%			\begin{cases}
%				c\cdot a_j  = \omega_j &\text{ if } j\in \{i_1,\dots,i_r\}\text{ and}\\
%				c \cdot a_j < \omega_j&\text{ if } j\notin \{i_1,\dots,i_r\}.\\
%			\end{cases}
%		\end{equation}
%		The subdivision $\cP(\omega)$ is called a \emph{regular triangulation} if all facets of the subdivision are simplices.
%		This happens if the vector $\omega$ is sufficiently generic.
%	\end{definition}
%	
%%	Geometrically, a regular subdivision arises by ``lifting'' the points in $\cP$ to $\R^{n+1}$ where we use the coordinates of $\omega$ as heights.
%%	The faces of $\cP(\omega)$ are then the ``lower faces'' of the convex hull of the lifted points, where a lower face is a face that has a normal vector with negative last coordinate.
%	
%	\begin{definition}
%		Let $\cR_n^0=\{e_i-e_j\mid i,j\in \left[n\right]\}$. This is  the set of vertices of the (full) \emph{root polytope} of type $A$ together with the origin.
%	\end{definition}
%	The next proposition establishes a relationship between regular subdivisions of  $\cR_n^0$ and KRW polytopes.
%	A similar statement also appeared in the second arXiv version of the preprint~\cite{JS19}.
%	
%	\begin{proposition} \label{RootTriangulations}
%		Let $\metric$ be an $n$-metric. 
%		\begin{enumerate}
%			\item The metric $\metric$ induces a regular subdivision $\cR_n^0(\metric)$ of  $\cR_n^0$ by assigning the height $\metric_{ij}$ to the points $e_i-e_j$ and the height $0$ to the origin.
%			\item The subdivision $\cR_n^0(\metric)$ is point-symmetric with respect to the origin and every maximal cell contains the origin.
%			\item For any index set $I\subseteq [n]^2$ the following are equivalent:
%			\begin{enumerate}
%				\item[(i)] The set $\{p_{ij} \mid (i,j) \in I\}$ is the set of all vertices contained in a facet of $\KRW{\metric}$.
%				\item[(ii)] The set $\{e_i-e_j \mid (i,j)\in I\}\cup\{0\}$ is a maximal cell of the regular subdivision $\cR_n^0(\metric)$.
%			\end{enumerate}
%		\end{enumerate}
%	\end{proposition}
%	
%	
%	
%%	\begin{proof} 
%%		The symmetry of the subdivision $\cR_n^0(\metric)$ follows immediately from the symmetry of the metric $\metric$.
%%		Moreover as the height of the origin is chosen to be $0$, the condition~\eqref{eq:subdivision} is satisfied for $a_j=0$ with an equality for any choice $c$.
%%		Hence, every facet of $\cR_n^0(\metric)$ contains the origin.
%%		
%%		Now fix $I\subseteq [n]^2$.
%%		The collection $\{p_{ij} \mid (i,j) \in I\}$ is the set of all $p_{ij}$  contained in a facet of $\KRW{\metric}$ if and only if 
%%		\[
%%		\textrm{there exists a vector }c\in \mathbb R^n
%%		\textrm{ with }
%%		\begin{cases}  c\cdot p_{ij}= 1 & \textrm{ if } (i,j)\in I \\
%%			c\cdot p_{ij} < 1 & \textrm{ if } (i,j)\not\in I.
%%		\end{cases}
%%		\]
%%		This is equivalent to requiring that
%%		\[
%%		\begin{cases}  c\cdot (e_i-e_j)= \metric_{ij} & \textrm{ if } (i,j)\in I \\
%%			c\cdot ( e_i-e_j  ) < \metric_{ij} & \textrm{ if } (i,j)\not\in I.
%%		\end{cases}
%%		\]
%%		So by definition of $\cR_n^0(\metric)$ (see \Cref{rem_dimW}) the latter condition is equivalent to the statement that the set $\{e_i-e_j \mid (i,j)\in I\}\cup\{0\}$ is a maximal cell of the regular subdivision $\cR_n^0(\metric)$.
%%	\end{proof}
%	
%	A fruitful construction of Gel'fand, Kapranov, and Zelevinsky encodes the various regular triangulations of a point configuration $\cP$ as points in a polyhedral fan called the {\em secondary fan} of $\cP$.
%	
%	\begin{theorem}[\cite{GKZ,DLRS}]\label{GKZ}
%		Let $\cP=\{a_1,\dots,a_k\}$ be  set of points in $\R^n$.
%		\begin{enumerate}
%			\item The \emph{secondary fan} $\Sigma(\cP)$ is a polyhedral fan in $\R^k$ such that two points $\omega,\omega'\in \R^k$ lie in the same open (secondary) cone of $\Sigma(\cP)$ if and only if the regular subdivisions $\cP(\omega)$ and $\cP(\omega')$ are equal.
%			\item The fan $\Sigma(\cP)$ is complete, that is the support of $\Sigma(\cP)$ is $\R^k$, as all points define some regular triangulation.
%			\item The subdivision $\cP(\omega)$ is a refinement of  $\cP(\omega')$ if and only if $\omega'$ is in the closure of the open cone that contains $\omega$.
%			\item The regular triangulations of $\cP$ correspond to the top-dimensional open cones of ~$\Sigma(\cP)$.
%		\end{enumerate}
%	\end{theorem}
%	
%	
%	
%	
%	\begin{remark}
%		\Cref{RootTriangulations} is the starting point for the study of KRW polytopes through symmetric subdivisions of $\cR_n^0$ where each maximal cell contains the origin.
%		In particular, one can enumerate the combinatorial types of KRW polytopes by enumerating the combinatorial types of such triangulations via the secondary fan of $\cR_n^0$ described above in \Cref{GKZ}.
%		This approach leads to the enumeration of the combinatorial types of KRW polytopes for the case $n=6$ using the \texttt{TOPCOM} software package~\cite{Rambau,Ram23}, see \Cref{sec:compuations}.
%	\end{remark}
%	
%	Building on \Cref{RootTriangulations} we can reconstruct the Wasserstein fan as a `symmetrized version' of the secondary fan of the root polytope:
%	
%	
%	\begin{proposition}
%		Let $\cR_n^0$ be as above.
%		Then, $\Sigma(\cR_n^0)$ is a fan in $\R^{n(n-1)+1}$, say with coordinates $x_{ij}$ for the vertices $e_i-e_j$ for $i\neq j\in \left[n\right]$ and $x_0$ for the origin.
%		Then
%		\[
%		\GPF{n}=\pi\big(\Sigma(\cR_n^0)\cap Q
%		\big)\cap \mcone{n},
%		\]
%		where $\pi:\R^{n(n-1)+1}\to \R^{\binom{n}{2}}$ is the projection sending $x_{ij}$ to $x_{\{i,j\}}$ for $1\le i<j\le n$, and $Q:=\{x\in \mathbb R^{n(n-1)+1}\mid x_{ij}=x_{ji}\textrm{ for }1\leq i<j\leq n \textrm{ and }x_0=0\}$.
%		
%		Moreover, the above equality is an equality of fans.
%	\end{proposition}
%%	\begin{proof}
%%		Let $\metric$ be an $n$-metric in $\mcone{n}$ and let $F_\metric$ be the unique open face of the Wasserstein fan $\GPF{n}$ that contains $\metric$.
%%		Furthermore, let $\metric'$ be another $n$-metric. By construction of the regular subdivisions (\Cref{RootTriangulations}.(1)) and by symmetry of $\metric$ and $\metric'$, surely $\cR_n^0(\metric)$ and $\cR_n^0(\metric')$ lie in $Q$.
%%		
%%		Now, by construction of $\GPF{n}$ and \Cref{thm:gordon_petrov}, the polytopes $\KRW{\metric}$ and $\KRW{\metric'}$ determine the same collection of admissible graphs and therefore have the same combinatorial structure if and only if $\metric'\in F_\metric$.
%%		By \Cref{RootTriangulations}, this means that $\cR_n^0(\metric)$ and $\cR_n^0(\metric')$ define the same regular subdivision of $\cR_n^0$ if and only if $\metric' \in F_\metric$. 
%%		Hence $\metric$ and $\metric'$ lie in the same open cone of $\Sigma(\cR_n^0)\cap Q$ if and only if $\metric'\in F_\metric$ which implies the claim.
%%	\end{proof}
	
	\section{Relationship to tight spans and injective hulls}\label{sec:tight_spans}
	
	We now want to relate the stratification of the metric cone by combinatorial type of KRW polytopes with the one in terms of the combinatorial type of a well-known polyhedral complex associated to metric spaces: the {\em injective hull} defined by Isbell \cite{Isbell} and rediscovered by Dress \cite{Dress}  with the name {\em tight span}. We start by outlining the construction and some key facts, and refer to \cite{Lan13} for a more detailed account and for a thorough analysis of the structure of injective hulls of general metric spaces.
	
	
	Given an $n$-metric $\metric$ we consider the (unbounded) polyhedron 
	$$P(\metric):=\{x\in \R^n \mid x_i + x_j \geq \metric_{ij} \textrm{ for all } i,j\in [n]\}.$$
	
	\begin{definition}[{\cite[Section 3]{Lan13}}, {\cite[Theorem 3.(v)]{Dress}}]\label{defIH}
		The injective hull, or tight span $E(\metric)$ of $\metric$ is the set of coordinate-wise minimal elements of $P(\metric)$. 
	\end{definition}
	
	\def\aff{\operatorname{aff}}
%	\begin{remark} Equivalently, $E(\metric)$ is the set of bounded faces of the polyhedron $P(\metric)$. This fact is stated in \cite[Lemma 1]{Dress89}. We sketch a proof following \cite[Section 4%
%		%p.\ 310
%		]{Lan13}. Notice that a point $p\in P(\rho)$ is coordinate-wise minimal if and only if for all $i\in [n]$ one of the defining inequalities containing $p_i$ is tight. That is, for all $i\in [n]$ there is $j\in [n]$ such that $p$ lies on the hyperplane $H_{ij}$ with equation $x_i+x_j=\rho_{ij}$.
%		Notice that every $H_{ij}$ is a supporting hyperplane of $P(\rho)$. Therefore $E(\rho)$ is the union of all faces $F$ of $P(\rho)$ such that there is $A\subseteq\binom{n}{2}$ with
%		\begin{equation}\label{alaLang}
%			\aff(F)= \bigcap_{\{i,j\}\in A} H_{ij} \textrm{ and }\cup A = [n], \end{equation}
%		i.e., every $i\in [n]$ is contained in some element of the index set $A$. These are bounded faces of $P(\rho)$: indeed, consider the vector $n_{ij}=e_i+e_j$, a normal vector to $H_{ij}$. Then $\aff(F)$ is perpendicular to a vector with all-positive coordinates (e.g., to the sum of all $n_{ij}$ for $\{i,j\}\in A$) and thus $F$ must be bounded because the intersection of $\aff(F)$ with the nonnegative orthant is. Conversely, every bounded face of $P(\rho)$ is of the form \eqref{alaLang}. By contraposition suppose that $\aff(F) = \bigcap_{\{i,j\}\in A} H_{ij}$ and $i_0\in [n]\setminus A$. Pick $p\in F$ and consider the half-ray $r:=p+e_{i_0} \mathbb R^+$.   Since $p\in P(\rho)$, obviously $r\subseteq P(\rho)$. Moreover $n_{ij}\perp e_{i_0}$, thus $r\subseteq H_{ij}$, whenever $i_0\not\in\{i,j\}$. Therefore $r\subseteq P(\rho)\cap \aff(F) = F$, and $F$ is not bounded.
%	\end{remark}
	
	Now recall the {\em second hypersimplex} 
	$\Delta_{2,n}:=\operatorname{conv}\{e_i+e_j \mid i,j\in[n],\, i\neq j\} \subseteq \R^{n}$.
	%
	%
	Following, e.g., \cite{HJ08}, every $n$-metric $\metric$ defines a lift of $\Delta_{2,n}$:
	$$
	L(\metric):=\operatorname{conv}\{(e_i+e_j, \metric_{ij}) \mid i,j\in [n], i\neq j\} + \mathbb R_{\geq 0}~ e_{n+1} \,\, \subseteq \R^{n+1}.
	$$
	The projection of the bounded faces of $L(\metric)$ onto $\R^n$ defines the regular subdivision  $\Delta_{2,n}(\metric)$. As was shown by Herrmann and Joswig \cite{HJ08}, there is a linear equivalence between $P(\metric)$ and the polar dual of the polyhedron $L(\metric)$ that induces a combinatorial isomorphism between the face poset of $E(\metric)$ and the face poset of~$\Delta_{2,n}(\metric)$.
	
	Thus, the combinatorial type of the tight span of an $n$-metric $\metric$ is dual to that of the induced regular subdivision of the second hypersimplex. In particular, the subdivision of the metric cone by combinatorial type of tight spans coincides with the intersection of the metric cone and the secondary fan of $\Delta_{2,n}$, the latter being a well-studied object in its own right (see, e.g., \cite{DST}). 
	
	
	
	
	
	
	
%	\begin{remark}[The case $n=4$]
%		The fan $\GPF{4}$ is a refinement of the subdivision of the metric cone induced by the  secondary fan of $\Delta_{2,4}$. See  \Cref{sec_n4}.
%	\end{remark}
	
	\begin{theorem} \label{thm:coarsefine}
		The fan $\GPF{n}$ is neither a coarsening nor a refinement of the subdivision of the metric cone induced by secondary fan of $\Delta_{2,n}$ for $n\geq 5$. 
	\end{theorem}
	
%	The proof of this theorem uses the following general fact that may be of independent interest. Recall that if $P$ and $Q$ are two polytopes in $\mathbb R^n$ both containing the origin and for which the affine hull of $P$ intersects the affine hull of $Q$ only in the origin, then the {\em free sum} of $P$ and $Q$ is $P\oplus Q:=\conv(P\cup Q)$, see for instance~\cite{Kal88}.
%	
%	\begin{lemma}\label{lem:directsum} Let $\metric_1$, $\metric_2$ be metrics on the sets % $X$, $Y$.
%		$[n]$ and $\{n,\ldots,n+m\}$, respectively. 
%		Consider the function
%		$
%		\metric_1\oplus \metric_2: [n+m]^2 \to \mathbb R
%		$
%		defined via 
%		$$
%		(\metric_1\oplus \metric_2)(i,j):=\begin{cases}
%			(\metric_1)_{ij} &
%			\text{if}\quad i,j\leq n, \\
%			(\metric_1)_{in}+(\metric_2)_{nj} & \text{if}\quad i\leq n < j, \\
%			(\metric_2)_{ij} & \text{if}\quad n\leq i,j.
%		\end{cases}
%		$$
%		Then $\metric_1\oplus \metric_2$ is a metric, and we have a direct (free) sum decomposition $$
%		\KRW{\metric_1\oplus \metric_2} = \KRW{\metric_1}\oplus \KRW{\metric_2}.
%		$$
%	\end{lemma}
%%	\begin{proof}
%%		A straightforward check of the definitions shows that $\metric_1\oplus\metric_2$ is a metric. 
%%		
%%		The set $X_1$ of pairs of distinct elements from $[n]$ is disjoint from the set $X_2$ of pairs of distinct elements of $\{n,\ldots,n+m\}$.
%%		
%%		The KRW polytopes of $\metric_1$ and $\metric_2$ both contain the origin and lie in the subspaces $\mathbb R^{X_1}$ and $\mathbb R^{X_2}$ of $\mathbb R^{\binom{n+m}{2}}$, respectively.
%%		Since the $X_i$ are disjoint, $\mathbb R^{X_1}\cap \mathbb R^{X_2} = \{0\}$. Then, by definition 
%%		$\KRW{\metric_1}\oplus \KRW{\metric_2}=\conv(\KRW{\metric_1}\cup \KRW{\metric_2}).$
%%		For brevity call $K$ the latter polytope, and let $\hat{\metric}:= \metric_1\oplus \metric_2$. 
%%		We have to prove that $K=\KRW{\hat{\metric}}$.
%%		
%%		By definition of $\hat{\metric}$, both $\KRW{\metric_1}$ and $\KRW{\metric_2}$ are contained in $\KRW{\hat{\metric}}$. Therefore, $K$  is contained in $\KRW{\hat{\metric}}$. The only potential vertices of $\KRW{\hat{\metric}}$ not contained in $K$ are given by vectors $\hat{p_{ij}}:=\frac{e_i-e_j}{\hat{\metric}_{ij}}$ with $i\leq n < j$.
%%		But by definition of $\hat{\metric}$ we have 
%%		$$
%%		\frac{e_i-e_j}{\hat{\metric}_{ij}} =
%%		\lambda\frac{e_i-e_n}{{\metric_1}_{in}}
%%		+
%%		\mu\frac{e_n-e_j}{{\metric_2}_{nj}}
%%		$$
%%		with $\lambda=\frac{({\metric_1})_{in}}{(\metric_1)_{in}+(\metric_2)_{nj}}$ and $\mu=\frac{({\metric_2})_{nj}}{(\metric_1)_{in}+(\metric_2)_{nj}}$ being positive numbers with $\lambda+\mu=1$. Therefore $\hat{p_{ij}}$ is in $K$ for all $i\leq n<j$, thus $\KRW{\hat{\metric}}=K$.
%%	\end{proof}
%	
%	Recall (e.g., from \cite[Item (1.4)]{Dress}) that, given an $n$-metric $\rho$ and any $i\in [n]$, the tight span $E(\rho)\subseteq \mathbb R^n$ contains the element $h_i$ defined as $(h_i)_j=\rho_{ij}$ for all $j\in[n]$. Moreover, $h_i$ is a vertex of the polyhedral complex $E(\rho)$.
%	
%	The following fact seems to be folklore but, since we did not find it explicitly stated in the literature, we include a direct proof based on \cite{BandeltDress} for the sake of self-containedness.
%	
%	\begin{lemma}\label{10zeilen} If $\rho$ is a metric on the set $[n]$ and $t$ a metric on the set $\{n,n+1\}$, then  the tight span of $\rho\oplus t$ is obtained from the tight span of $\rho$ by gluing a new edge at the vertex given by $h_n$.
%	\end{lemma}
%	
%	
%	
%%	\begin{proof} Write  $\lambda:=t_{n\, n+1}=(\rho\oplus t)_{n\, n+1}$, and consider the pseudometrics $\sigma$ and $\delta$ on $[n+1]$ where $\sigma$ is given by
%%		$\sigma_{ij}:=\rho_{ij} $ for $j\neq n+1$, $\sigma_{i\, n+1}:=\rho_{in}$, and  $\delta:= \lambda\cdot\delta_{[n],\{n+1\}}$ is a (nonzero) multiple of the split pseudometric defined in \Cref{defsplits}. As in \cite{BandeltDress}, we can apply the construction of the tight span also to pseudometrics, obtaining
%%		\begin{align*}
%%			E(\sigma) &=\{x\in\mathbb R^{[n+1]} \mid (x_1,\ldots,x_n)\in E(\rho), x_{n+1}=x_n\},\\ 
%%			E(\delta) &=\{x\in\mathbb R^{[n+1]} \mid x_1=x_2=\ldots=x_n,(x_n,x_{n+1})\in E(t)\}\\ 
%%			&= \{(\alpha,\ldots,\alpha,\lambda-\alpha)\in\mathbb R^{n+1} \mid 0\leq \alpha\leq \lambda\},
%%		\end{align*}
%%		which are  polyhedral complexes isomorphic to $E(\rho)$ and $E(t)$, respectively. Let $$\hat{h}_n :=(\rho_{in},\ldots,\rho_{nn},\rho_{nn})$$ denote the image of $h_n\in E(\rho)$ in $E(\sigma)$, and 
%%		let\footnote{Sum of subsets of Euclidean space is intended elementwise, i.e., as Minkowski sum.} 
%%		\def\hE{\hat{E}}
%%		$$\hE(\sigma) := E(\sigma) + (0,0,\ldots,0,\lambda),\quad
%%		\hE(\delta):= E(\delta) + \hat{h}_n.$$
%%		
%%		Since $\hE(\sigma) \cap \hE(\delta)=\{\hat{h}_n\}$, it will be enough to show that
%%		\begin{equation}\label{almostlast}
%%			E(\rho\oplus t) = \hE(\sigma) \cup \hE(\delta),
%%		\end{equation}
%%		since $\hE(\delta)$ (isometric to the unit interval) is joined to $\hE(\sigma)$ (isomorphic to $E(\rho)$) at the unique common point $\hat{h}_n + (0,\ldots,0,\lambda)$, the image of $h_n$ in $E(\rho)$.
%%		
%%		The inclusion $E(\rho\oplus t) \supseteq \hE(\sigma) \cup \hE(\delta)$ can either be checked directly with \Cref{defIH} or deduced from the fact that $\hE(\sigma)$, resp.\ $\hE(\delta)$ are the images of $E(\rho)$, resp.\ $E(t)$, under the canonical inclusion into $E(\rho\oplus t)$ described in \cite[Item (1.11)]{Dress}\footnote{In the language of \cite[Item (1.11)]{Dress}, we are considering the inclusions of tight spans $\tau_{[n]}$, resp.\ $\tau_{n+1}$, induced by the extensions of $\rho$ to $\rho\oplus t$, resp.\ of $t$ to $\rho\oplus t$.}.
%%		
%%		For the other inclusion start by noticing that, obviously, $\rho\oplus t = \sigma + \delta$. The isolation index of $n+1$ with respect to $\rho\oplus t$ (see \Cref{def:ii} below for the definition) is   $\alpha_{[n],\{n+1\}}^{\rho\oplus t} = \lambda$. Therefore \cite[Theorem 7]{BandeltDress} applies and shows $P(\rho\oplus t) = P(\sigma)+P(\delta)$. As is explained, e.g., in \cite[p. 89]{BandeltDress}\footnote{In \cite{BandeltDress} our $E(\rho)$ is denoted by $T(\rho)$.}, this implies $E(\rho\oplus t)\subseteq E(\sigma) + E(\delta)$.  Now for the inclusion $E(\rho\oplus t) \subseteq \hE(\sigma) \cup \hE(\delta)$ it is enough to show that 
%%		\begin{equation}\label{atlast}
%%			E(\rho\oplus t) \cap (E(\sigma) + E(\delta)) \subseteq 
%%			\hE(\sigma) \cup \hE(\delta).
%%		\end{equation}
%%		Take $p\in (E(\sigma) + E(\delta))$. In our setup, without loss of generality we can write
%%		$$
%%		p= (y_1+\alpha,\ldots, y_n+\alpha, y_n + (\lambda - \alpha)),\quad \textrm{ with }
%%		y\in E(\rho), \;\; 0\leq \alpha\leq \lambda.
%%		$$
%%		We surely have that $p\in P(\rho\oplus t)$, since:
%%		
%%		\begin{itemize}
%%			\item[(i)] for $i,j\in [n]$: $p_i + p_j = y_i+ y_j + 2\alpha \geq \rho_{ij} + 2\alpha \geq (\rho\oplus t)_{ij} +2\alpha \geq (\rho\oplus t)_{ij}$;
%%			\item[(ii)] for $i\in [n]$: $p_i+p_{n+1} = y_i + \alpha + y_n + \lambda - \alpha = y_i + y_n +\lambda \geq \rho_{in} + t_{nn+1} = (\rho\oplus t)_{i,n+1}$.
%%		\end{itemize}
%%		
%%		
%%		If $\alpha = 0$, then $y\in E(\rho)$  implies that $(p_1,\ldots,p_n)$ is already coordinate-wise minimal satisfying  (i). In particular, there is $i_0\leq n$ with $y_{i_0}+y_n = \rho_{{i_0}n}$. This implies that $p_{n+1}$ cannot be made smaller either, because $p_{{i_0}}+p_{n+1}=y_{i_0} + y_n +\lambda = \rho_{{i_0}n}+t_{n n+1} = (\rho + t)_{{i_0} n+1} $ is an equality. In this case, then, $p\in E(\rho\oplus t)$ implies $p\in \hE(\sigma)$.
%%		
%%		If $\alpha > 0$, then $y\in E(\rho)$  implies that none of the inequalities in (i) is an equality, indeed $p_i+p_j\geq \rho_{ij}+2\alpha$ for all $i,j\in [n]$. 
%%		
%%		Now if $y_j + y_n > \rho_{in}$ for some $j$, define $\tilde{p}$ by $\tilde{p}_j:=p_j-{\alpha}$ and $\tilde{p}_i:=p_i$ for $i\neq j$. Then $\widetilde{p}\in P(\rho\oplus t)$, yet $\tilde{p}$ is smaller than $p$ in the $j$-th coordinate. Thus $p\in E(\rho\oplus t)$ implies that $y_j + y_n = \rho_{in}$ for all $j\in [n]$, so $p\in \hE(\delta)$. This concludes the proof of \Cref{atlast}, hence of \Cref{almostlast} and, with it, of the claim.
%%	\end{proof}
%	
%	\begin{definition}\label{def:tk}
%		For given $k>0$ let $\tk{n}{k}$ denote the $k$-metric defined on the set $\{n,\ldots,n+k\}$ by setting $t^{(k)}_{i,j} := \vert j-i\vert$. This is a graph metric realized on the full set of vertices of a path with $k+1$ vertices.
%	\end{definition}
%	
%	\begin{remark}\label{appendpath} 
%		Let $\rho$ be any $n$-metric. Then $$\rho\oplus \tk{n}{k}=
%		(\cdots (\rho\oplus \tk{n}{1}) \oplus \tk{n+1}{k})\cdots )\oplus \tk{n+k-1}{1}.
%		$$
%		By repeated application of \Cref{10zeilen} we obtain that the tight span of $\rho\oplus \tk{n}{k}$ is obtained from the tight span of $\rho$ by appending a $k+1$-path at the vertex $h_n$ of $E(\rho)$.
%	\end{remark}
%	
%%	\begin{proof}[Proof of \Cref{thm:coarsefine}]
%%		\Cref{example_decomp}  exhibits two $5$-metrics that lie in the same open cone of~
%%		$\CA{5}$ (thus they have combinatorially isomorphic KRW polytopes), but lie in different cones of the secondary fan of $\Delta_{2,5}$ (and thus have different tight spans). 
%%		Moreover, \Cref{ex:4pt} shows two metrics on $4$ points that are in the same cone of the secondary fan of $\Delta_{2,4}$ but have non-isomorphic KRW polytopes. %\emanuele{Can we, from these, counterexamples for every $n\geq 5$? Idea:}
%%		Starting from those, we are going to construct examples for any (finite) number of points.
%%		
%%		Fix $k>0$ and recall the metrics  $\tk{n}{k}$ defined in \Cref{def:tk}. Since these are graph metrics realized on the full set of vertices of a tree with $k+1$ vertices, $\KRW{\tk{n}{k}}$ is unimodularly equivalent to the cross-polytope of dimension $k$. Moreover, the tight span of $t^{(k)}$ is combinatorially equivalent to the tree realizing it, i.e., a path with $k$ edges, with vertices labeled $n,\ldots,n+k$ (see, e.g., \cite[Theorem 8]{Dress}).
%%		
%%		Now given an $n$-metric $\metric$ consider the $n+k$-metric $\metric^{(k)}:=\metric\oplus \tk{n}{k}$. 
%%		
%%		By 
%%		
%%		\Cref{appendpath}, the 
%%		tight span $E(\metric^{(k)})$ is obtained from that of $\metric$ by appending a $k$-path at the vertex labeled $n$.
%%		
%%		Moreover, by \Cref{lem:directsum} there is a direct (free) sum decomposition $\KRW{\metric^{(k)}}=\KRW{\metric}\oplus \KRW{t^{(k)}}$, implying that the poset of non-maximal faces of $\KRW{\metric^{(k)}}$ is the Cartesian product of the posets of non-maximal faces of $\KRW{\metric}$ and of $\KRW{t^{(k)}}$ (see, e.g., \cite{Kal88}).
%%		
%%		Let $\metric_1$, $\metric_2$ be the two $5$-metrics of \Cref{example_decomp}. The isomorphism between  the polytopes $\KRW{\metric_1}$ and $\KRW{\metric_2}$ induces an isomorphism between $\KRW{\metric_1^{(k)}}$ and $\KRW{\metric_2^{(k)}}$. On the other hand, any isomorphism between the tight spans of $\metric_1^{(k)}$ and $\metric_2^{(k)}$ must map the $t^{(n,k)}$-part to itself (since the tight spans of $\metric_1$ and $\metric_2$ do not contain any path of length more than $1$ consisting of vertices with degree $\leq 2$, see \Cref{example_decomp}), and thus would induce an isomorphism between the tight spans of $\metric_1$ and $\metric_2$. The latter cannot exist, and so we show that $\GPF{n}$ is not a refinement of the secondary fan of $\Delta_{2,n}$ for all $n\geq 5$.
%%		
%%		Let $\metric_1$, $\metric_2$ be the two $4$-metrics of \Cref{ex:4pt}. The combinatorial isomorphism between the tight span of $\metric_1$ and $\metric_2$ extends to an isomorphism of the tight spans of $\metric_1^{(k)}$ and $\metric_2^{(k)}$. On the other hand the fact that $\KRW{\metric_2}$ is simplicial implies simpliciality of $\KRW{\metric_2^{(k)}}$, and since $\KRW{\metric_1}$ is not simplicial, neither is $\KRW{\metric_1^{(k)}}$. In particular, $\KRW{\metric_1^{(k)}}$ is not isomorphic to $\KRW{\metric_2^{(k)}}$ for any $k>0$. We have proved that for all $n\geq 5$ the fan $\GPF{n}$ is not a coarsening of the secondary fan of $\Delta_{2,n}$.
%%	\end{proof}
%	
%	
%	
%	\begin{example}\label{ex:4pt}
%		Consider the following two $4$-metrics.
%		$$
%		\metric_1:=\begin{pmatrix}
%			0 & 3 & 3 & 4 \\
%			3 & 0 & 4 & 3 \\
%			3 & 4 & 0 & 3 \\
%			4 & 3 & 3 & 0
%		\end{pmatrix}
%		\quad\quad
%		\metric_2:=\begin{pmatrix}
%			0 & 4 & 3 & 5 \\
%			4 & 0 & 5 & 3 \\
%			3 & 5 & 0 & 4 \\
%			5 & 3 & 4 & 0
%		\end{pmatrix}
%		$$
%		These metrics have combinatorially isomorphic tight spans, but non-isomorphic KRW polytopes, see \Cref{fig:4pt}.
%		The polytopes are not combinatorially isomorphic as $\KRW{d_1}$ has two square faces whereas $\KRW{d_2}$ is simplicial.
%	\end{example}
%	
%	
%	
%	\begin{figure}[ht]
%		\centering
%		\begin{subfigure}[t]{0.3\textwidth}
%			\centering
%			\resizebox{\textwidth}{!}{\input{figures/krw_rho1.tikz}}
%			\caption{$\KRW{d_1}$}
%			\label{fig:4pta}
%		\end{subfigure}
%		\begin{subfigure}[t]{0.3\textwidth}
%			\centering
%			\includegraphics[width=.7\textwidth]{figures/TS_12}
%			\caption{tight span of both metrics}
%			\label{fig:4ptb}
%		\end{subfigure}
%		\begin{subfigure}[t]{0.3\textwidth}
%			\centering
%			\resizebox{\textwidth}{!}{\input{figures/krw_rho2.tikz}}
%			\caption{$\KRW{d_2}$}
%			\label{fig:4ptc}
%		\end{subfigure}
%		\caption{Illustration for \Cref{ex:4pt}. (a), (c): KRW polytopes of the metrics $\metric_1$, $\metric_2$. (b): Combinatorial type of the tight span. The horizontal edges are of length $1$ and $2$ for the metrics $\metric_1$ and $\metric_2$, resp. All other edges are of length $1$.}
%		\label{fig:4pt}
%	\end{figure}
%	
%	
%	For an example of two metrics with isomorphic polytopes, but different tight spans, see \Cref{example_decomp}. 
	
	%\section{Noteworthy subfans}\label{sec:subfans}
%	\subsection{Tree-like metrics} An $n$-metric $\metric$ is called {\em tree-like} if it can be represented as the shortest-path metric among a subset of $n$ vertices of a weighted tree $T(\metric)$. It is known since work of Buneman \cite{Buneman} that an $n$-metric $\metric$ is tree-like if and only if it satisfies the four-point condition 
%	\begin{equation}\label{tree-condition}
%		\metric_{ij} + \metric_{kl} \leq \max\{\metric_{ik} + \metric_{jl}, \metric_{il}+\metric_{jk}\} 
%		\textrm{ for all } i,j,k,l\in [n].
%	\end{equation}
%	
%	These are exactly the metric spaces whose tight span is a tree (i.e., a one-dimensional contractible polyhedral complex), see e.g., \cite[Theorem 8]{Dress}.
%	
%	\begin{proposition}\label{prop:tl}
%		The set of tree-like metrics is a subfan of $\GPF{n}$. In particular, the set of phylogenetic trees on $n$ leaves is a union of (relatively open) cells of $\GPF{n}$.
%	\end{proposition}
%%	\begin{proof} A metric $\metric$ satisfies the four-point condition \eqref{tree-condition} for a quadruple $A=\{i,j,k,l\}\in \binom{n}{4}$ if and only if, of the three numbers $\metric_{ij} + \metric_{kl},\metric_{ik} + \metric_{jl}, \metric_{il}+\metric_{jk}$, two are equal and the third does not exceed the other two. In turn, this is equivalent to saying that there is a permutation $\pi_A=(i',j',k',l')$ of the elements of $A$ such that $$\metric\in \mathscr C_{\pi_A}:= H_{(i',j'),(k',l')} \cap H_{(i',k'),(j',l')}^+.$$
%%		This means that a metric $\metric$ is tree-like if and only if there is a choice of a permutation $\pi_A$ for every $A\in \binom{n}{4}$ such that $\metric \in \bigcap_{A\in \binom{n}{4}} \mathscr C_{\pi_A}$. Thus, letting $S_A$ denote the group of permutations of the set $A$, the set of tree-like metrics can be expressed as 
%%		$$
%%		\bigcap_{A\in \binom{n}{4}} 
%%		\bigcup_{\pi_A \in S_A
%%		}
%%		\mathscr C_{\pi_A}.
%%		$$
%%		Since each  $\mathscr C_{\pi_A}$ is a subfan of~$\GPF{n}$, it follows that the set of all tree-like metrics is a subfan of~$\GPF{n}$ as well.
%%	\end{proof}
%	
%	
%	
%	
%	
%	
%	
%	
%	
%	\subsection{Circular-decomposable metrics}
%	
%	An $n$-metric $\metric$ is circular-decomposable if there is a cyclic ordering $\prec$ of $[n]$ such that 
%	\begin{equation}\label{circular_condition}
%		\metric_{ij} + \metric_{kl} \leq \metric_{ik} + \metric_{jl} \textrm{ whenever } i\prec j \prec k \prec l.
%	\end{equation}
%	To our knowledge this condition first appeared in work of Kalmanson \cite{kalmanson}, and was proved in \cite{CF98} to be equivalent to the ``maximal compatibility'' condition studied by Bandelt and Dress in \cite{BandeltDress}. 
%	
%	From \Cref{circular_condition} it follows immediately that a point in $\R^{\binom{n}{2}}$ corresponds to a circular-decomposable (pseudo)metric if and only if it is contained in the cone
%	\begin{equation}\label{eq:Kcirc}
%		K_\prec:=\bigcap_{i\prec j \prec k \prec l} H^+_{(i,l),(j,k)}
%		\textrm{ for some circular ordering }\prec \textrm{ of } [n].
%	\end{equation}
%	
%	The following definition is from \cite{Devadoss-petti}. 
%	
%	\begin{definition}
%		A split $A\uplus B = [n]$ is said to be compatible with a circular ordering $\prec$ if $i\prec j$ for all $i\in A$
%		and all $j\in B$. Equivalently, we can label the elements of $[n]$ as $x_1,\ldots,x_n$ in such a way that $x_i\prec x_j$ if $i\in A $ and $j\in B$.
%	\end{definition}
%	
%	\begin{lemma} \label{lem:circular}
%		For every cyclic ordering $\prec$ the set $K_{\prec}$ is a simplicial cone of dimension ${\binom{n}{2}}$. Its rays are generated by the pseudometrics associated to the splits that are compatible with $\prec$.  
%		Irrespective of the chosen $\prec$, the cone $K_{\prec}$ contains the cone $\elsplitsclosed{n}$ generated by the elementary splits.
%		%The wall opposite the ray generated by the split $\{x_i\prec \ldots \prec x_{j-1}\}\cup\{x_j\prec\ldots\prec x_{i-1}\}$ is $H_{(i-1, j),(i,j-1)}$.
%	\end{lemma}
%%	
%%	\begin{proof}
%%		The last claim follows after noticing that elementary splits are compatible with every circular ordering. The other claims are proved as items (i-iii) in \cite[Theorem 5, p.\ 75]{BandeltDress}.
%%	\end{proof}
%	
%	
%	
%	\begin{theorem}\label{thm:cd}
%		The set of all circular decomposable pseudometrics on $n$ points is the full-dimensional simplicial subfan of $\GPFclosed{n}$ given by all points contained in
%		$$
%		\kfanclosed{n}:=\bigcup_{\substack{\prec \textrm{ circular}\\ \textrm{ordering of }[n]}} K_\prec. 
%		$$
%		Accordingly, the circular decomposable metrics form a full-dimensional subfan $\kfan{n}$ of $\GPF{n}$.
%	\end{theorem}
%%	\begin{proof} \Cref{eq:Kcirc} implies immediately that $\kfanclosed{n}$ is indeed a subfan of $\GPFclosed{n}$. The claim about simpliciality and dimension follows from \Cref{lem:circular}.
%%	\end{proof}
%%	
%%	\begin{remark}[The space of Circular Split Networks]\label{DP-HKT}
%%	A metric is circular decomposable if and only if it can be realized by a {\em
%%		circular split network} \cite[Section 2.2]{Devadoss-petti}. Devadoss and Petti
%%	\cite{Devadoss-petti} define and study a simplicial fan $\operatorname{CSN}_n$
%%	that parametrizes all circular split networks on $n$ points associated to
%%	circular decomposable metrics without trivial splits. 
%%\end{remark}
%%
%%
%%\begin{corollary} The set of circular decomposable metrics without trivial
%%	splits defines a subfan of $\GPFclosed{n}$ consisting of all cones of
%%	$\kfanclosed{n}$ whose closure does not contain any ray of $\elsplitsclosed{n}$.
%%	This fan structure on the set of circular decomposable metrics without trivial
%%	splits is combinatorially isomorphic to the one defined and studied in
%%	\cite{Devadoss-petti} and \cite{HKT}, see \Cref{DP-HKT} above.
%%\end{corollary}
%	
%	
%	
%	
%	
%	
	
	
	
	
	\subsection{Split decomposition} %\leonie{aus Delucchi-Hoessly und Bandelt-Dress} 
	We now turn to the class of totally split-decomposable spaces, as introduced by Dress and studied by Bandelt and Dress \cite{BandeltDress}.
	\begin{definition}\label{def:ii}
		Let $\metric$ be a symmetric function on a set $X$ and let $A, B \subseteq X$. Define the \emph{isolation index} $\alpha_{A,B}^\metric$ 
		as \[\alpha_{A, B}^\metric = \frac{1}{2} \cdot \min_{\substack{a,a'\in A \\b,b'\in B}} \left(\max\{a'b+b'a, a'b'+ ab, aa'+bb'\}-(aa'+bb')\right), \] where $uv = \metric(u,v)$. 
	\end{definition}
	
	\begin{remark}
		As noted in \cite{BandeltDress}, $\alpha^\rho_{A,B} \geq 0$, $\alpha^\rho_{A,B} =0$ whenever $A\cap B \not= \emptyset$. Moreover, if $\metric$ is a metric and both $A$ and $B$ have at least 2 elements, then  $\alpha^\rho_{A,B} = \alpha^\rho_{\{a,a'\}, \{b,b'\}}$ for some points $a, a'\in A$, $b, b' \in B$ with $a\neq a'$, $b\neq b'$. 
		
		In the following, when writing expressions such as $\alpha^\rho_{\{a,a'\}, \{b,b'\}}$
		we will assume that $a,a',b,b'$ are pairwise distinct points.
	\end{remark}
	
	A bipartition $A \uplus B=[n]$ is called a $\metric$-split if $\alpha^\metric_{A,B} > 0$. 
	A symmetric function $\rho$ which does not admit any $\rho$-splits is called \emph{split-prime}. 
	\begin{theorem}\label{thm:dec}[\cite[Theorem 2 and Corollary 2]{BandeltDress}]
		Every symmetric function $\metric: X \times X\rightarrow \mathbb{R}$ on a finite set $X$ can be expressed as \[ \metric = \metric_0 + \sum \alpha^\metric_{A,B} \cdot \delta_{A,B}\] with $\metric_0$ split-prime. The metric $\metric_0$ is uniquely determined.
	\end{theorem}
	
	\begin{definition}\label{def:cd}
		A metric $\metric$ is called \emph{ totally split-decomposable} if it can be written as a positively weighted sum of split metrics. Equivalently, $\metric_0 = 0$ in the above expression. 
	\end{definition}
	
	\noindent The following is a criterion for checking whether a metric is totally split-decomposable:
	\begin{theorem}[{\cite[Theorem 6]{BandeltDress}}] 
		\label{decomptheo}
		Let $\metric: X \times X \rightarrow \mathbb{R}$ be a symmetric function with zero diagonal. 
		
		The following conditions are equivalent: 
		\begin{enumerate}
			\item $\metric$ is a totally split-decomposable metric. 
			\item For all $t,u,v,w,x \in X$ it holds that $\alpha_{\{t,u\}, \{v,w\}} \leq \alpha_{\{t,x\}, \{v,w\}} + \alpha_{\{t,u\}, \{v,x\}}$.
		\end{enumerate}
	\end{theorem}
	
	\begin{remark}
		From \Cref{decomptheo} it is  clear that all metrics on $4$ points are split-de\-com\-pos\-able. For $\vert X \vert \geq 5$ we can ask the question whether two metrics in the same cone of the Wasserstein arrangement are either both split-decomposable or both not split-decomposable. The next example answers this question in the negative.
	\end{remark}
	
	\begin{example}\label{example_decomp} 
		Consider the two metrics $(\metric_1, \metric_2)$ on five points given by the following matrices:
		
		\begingroup
		\[\metric_1 = \begin{pmatrix} 0 & 125 & 48 & 149 & 84 \\ 125 & 0 & 149 & 48 & 99  \\48 & 149 & 0 & 125 & 77 \\  149 & 48 & 125 & 0 & 92 \\ 84 & 99 & 77 & 92 & 0 \end{pmatrix},~~~\metric_2 = \begin{pmatrix} 0 & 7447 & 4316 & 10083 & 5584 \\ 7447 & 0 &  10083 & 4316 & 5199 \\ 4316 & 10083 & 0 & 7447 & 7560 \\ 10083  & 4316 & 7447 & 0 & 6179 \\ 5584 & 5199 & 7560 & 6179 & 0 \end{pmatrix}. \]
		\endgroup 
		These metrics lie  in the same cone of the Wasserstein arrangement $\CA{5}$, and therefore the associated KRW polytopes have the same combinatorial structure.
		%The KRW polytope of $\metric_1$ is depicted in \Cref{fig:schlegel}. Sketches of the tight spans are depicted in Figures \ref{fig:tight_span_rho_1} and \ref{fig:tight_span_rho_2}. 
		
		The metric $\metric_1$ is split-decomposable but for the metric $\metric_2$ we find \[\alpha^{\metric_2}_{\{5,2\}, \{1,3\}} > \alpha^{\metric_2}_{\{2,4\}, \{1,3\}} + \alpha^{\metric_2}_{\{5,2\}, \{1,4\}}.\] 
		By \Cref{decomptheo}, this certifies  that $\metric_2$ is not split-decomposable, which in turn implies that the split-decomposable metrics do not form a subfan of the Wasserstein fan.
		
		\begin{example} 
		Consider the following two $4$-metrics.
		$$
		\metric_1:=\begin{pmatrix}
			0 & 3 & 3 & 4 \\
			3 & 0 & 4 & 3 \\
			3 & 4 & 0 & 3 \\
			4 & 3 & 3 & 0
		\end{pmatrix}
		\quad\quad
		\metric_2:=\begin{pmatrix}
			0 & 4 & 3 & 5 \\
			4 & 0 & 5 & 3 \\
			3 & 5 & 0 & 4 \\
			5 & 3 & 4 & 0
		\end{pmatrix}
		$$
		These metrics have combinatorially isomorphic tight spans, but non-isomorphic KRW polytopes.
		The polytopes are not combinatorially isomorphic as $\KRW{d_1}$ has two square faces whereas $\KRW{d_2}$ is simplicial.
	\end{example}
%		\begin{figure}[h!]
%			\centering 
%			\includegraphics[scale=.44]{figures/schlegel_dual}
%			\caption{Schlegel diagram of the polar dual of the KRW polytope (i.e., the Lipschitz polytope) of the metric $\metric_1$ from  \Cref{example_decomp}.}
%			\label{fig:schlegel}
%		\end{figure}
		
%		\begin{figure}
%			\includegraphics[width=0.8\textwidth]{figures/Zeichnung_tight_span_rho_1.pdf} 
%			\caption{Sketch of the tight span of the metric $\rho_1$. It is a special case of type II as established in \cite{BandeltDress} with encoding of the vertices as $(1,2,3,4,5) = (z_1, z_2, y_1, y_2, y_3)$. 
%				The values of the parameters are as follows: $(\zeta_1, \zeta_2, \eta_1, \eta_2, \eta_3, \beta_1, \beta_2, \beta_3, \beta_4, \alpha) = (12,12,12,12,13.5, 43, 15.5, 58, 8.5, 0)$. This metric is not generic in the sense of \cite{BandeltDress}, since one of the parameters ($\alpha$) is zero.}
%			\label{fig:tight_span_rho_1}
%			
%			\includegraphics[width=0.8\textwidth]{figures/Zeichnung_tight_span_rho_2.pdf} 
%			\caption{Sketch of the tight span of the metric $\rho_2$. It is also of type II  with the encoding of the vertices as $(1,2,3,4,5) = (z_1, z_2, y_1, y_2, y_3)$. The values of the parameters are as follows: $(\zeta_1, \zeta_2, \eta_1, \eta_2, \eta_3, \beta_1, \beta_2, \beta_3, \beta_4, \alpha) = (840,840,840,840, 840, 3076, 330, 2193, 1808, 498)$. }
%			\label{fig:tight_span_rho_2}
%		\end{figure}
		
	\end{example}
%	\begin{remark}
%		Note that the notion of ``generic'' metric discussed in  \cite{BandeltDress} is not the same as the one of \Cref{def:KRW}. In \cite{BandeltDress}, this term is used in contrast to ``degenerate'' metrics, where some coefficients $\alpha^\metric_{A,B}$ in the expression of \Cref{thm:dec}  might have value 0. 
%	\end{remark}
	
%	In general, taking a look at the condition for decomposability for $n = 5$, since the isolation indices are defined on sets of two elements, the minimum runs just over one possibility. 
%	
%	For the maximum, this is determined by the chamber of $\W_5$ the metric is in: The maximum over the three values can be computed by pairwise comparison, but comparing the values $ab+a'b'$ and $a'b+ab'$ is nothing other than asking which of the two edge pairs $\{(a,b), (a',b')\}, \{(a',b), (a,b')\}$ is admissible. 
%	
%	
%	Thus, restricting ourselves to the metrics of a specific cone of the Wasserstein Arrangement for $n=5$, the conditions of \Cref{decomptheo} form a local hyperplane  arrangement within this cone where the split-decomposable metrics lie in exactly one chamber. 
	
	
	
%	\subsection{Consistent totally split-decomposable metrics} 
%	
%	Let $\metric$ be a totally split-decomposable metric. In \cite[Theorem 1]{SixPoints} (see also \cite{AnExplicit})  it is proved that the tight span of $\metric$ is combinatorially isomorphic to the so-called {\em Buneman complex} if and only if $\metric$ satisfies the following \emph{six-point condition}:\\
%	
%	\noindent{\eqnum\label{def:SixPoints}}
%	\begin{minipage}[t]{.9\linewidth}
%		For all $I\subseteq [n]$ with $\vert I \vert = 6$ there are distinct elements $i,j\in I$ with
%		\begin{equation*}
%			\metric_{ij} + \metric_{kl} \leq \max\{\metric_{ik}+\metric_{jl},\metric_{jk}+\metric_{il}\}
%		\end{equation*}
%		for all pairs of distinct $k,l \in I\setminus\{i,j\}$.
%	\end{minipage}
%	
%	
%	
%	
%	\begin{proposition}\label{prop:6p}
%		The set of metrics that satisfy the six-point condition \Cref{def:SixPoints} is a subfan of $\GPFclosed{n}$.
%	\end{proposition}
%	\begin{proof}
%		It is enough to prove that the set of all $\metric\in \mathbb R^{{\binom{n}{2}}}$ satisfying the six-point condition is a union of faces of the Wasserstein arrangement. For this, notice that the inequality in \Cref{def:SixPoints} is satisfied whenever 
%		$\rho\in T_{ijkl}$, where
%		$$
%		T_{ijkl}=H^+_{(i,k),(j,l)} \cup H^+_{(i,l),(j,k)}
%		$$
%		is a union of (closed) chambers of $\CA{n}$. Then, $\metric$ satisfies  the six-point condition in full if and only if
%		$$
%		\metric \in \bigcap_{
%			\substack{I\subseteq [n]\\
%				\vert I \vert = 6}}
%		\;\;
%		\bigcup_{
%			\substack{i,j\in I \\
%				i\neq j }}
%		\;\;
%		\bigcap_{
%			\substack{k,l\in I\setminus \{i,j\}\\
%				k\neq l
%		}}
%		T_{ijkl}.
%		$$
%		This proves that the set of $\metric$ that satisfy the six-point condition is obtained uniting and intersecting sets of (closed) chambers, hence it is a union of faces of the arrangement $\GPFclosed{n}$. 
%	\end{proof}
%	
%	\begin{remark} \label{rem:consistent}
%		A totally split-decomposable metric that satisfies \eqref{def:SixPoints} is called {\em consistent} in \cite{SixPoints}. We note that the set of consistent totally split-decomposable metrics is not a subfan of the Wasserstein fan. 
		
		
%		To prove this, let $\metric_1$, $\metric_2$ be the $5$-metrics defined in \Cref{example_decomp}, let $\tk{5}{2}$ the ``path metric on two edges'' defined in \Cref{def:tk}, and recall the ``direct sum'' construction of \Cref{lem:directsum}. 
%		The proof of \Cref{thm:coarsefine} shows that the two $7$-metrics
%		$$
%		\rho_1 \oplus (12 \cdot \tk{5}{2}),\quad\quad \rho_2 \oplus (840 \cdot \tk{5}{2}),
%		$$
%		are in the same cone of the Wasserstein fan $\W_7$, and a computer check reveals that both satisfy the six-point condition \eqref{def:SixPoints}. However, the first metric is totally split-decomposable, while the second is not. (This can be either checked by computer or proved  noticing that for every $n$-metric $\rho$ the metric $\rho\oplus (\lambda \cdot \tk{n}{k})$ decomposes as the sum of $\rho$ and $k$ elementary splits, and using the fact that $\rho_1$ is split-decomposable but $\rho_2$ is not, see \Cref{example_decomp}.) This shows that  consistent totally split-decomposable metrics do not form a subfan of~$\GPFclosed{n}$.
%	\end{remark}
	
	
	
	
	
	\section{The \texorpdfstring{$f$}{f}-vector in the strict case}\label{sec:strict_case}
	
	In this section, we recall results for the number of faces of KRW-polytopes in dimensions 0 and 1 and give a formula for the number of 2-dimensional faces. 
	
	The starting point of this section is the following result by Gordon and Petrov. Recall from \Cref{def:KRW} the definition of a generic metric.
	
	\begin{theorem}[{\cite[Theorem 1]{GordonPetrov}}]
		Let $X, \vert X \vert = n+1$ be a metric space with generic metric $\metric$. Then for $0 \leq m \leq n$, the number of $(n-m)$-dimensional faces of the Lipschitz polytope of $\metric$ is \[\binom{n + m}{m,m,n-m}.\]  
	\end{theorem}
	
	
	%The proof of this theorem uses the {\em admissible graphs} discussed earlier in \Cref{sec:GP}. 
	%In fact, using \cite[Corollary 1.(3)]{GordonPetrov} the number of faces of dimension $(n-m)$ in the generic case agrees with the number of different admissible (directed) forests with $m$ edges on $n+1$ vertices. 
	
	
	The strict, non generic case is somewhat more complicated, but general formulae for faces of dimension $0$ and $1$ can be stated by using the graphic characterization of the faces.  %A set of vertices $F$ of the polytope defines a face if the intersection of all facets containing $F$ is identical to $F$. In order to check whether a given (admissible) graph $G$ is the graph $G(F)$ for some a face $F$ of the polytope (see \Cref{sec:GP}), we need to check that for every edge $e$ not in $G$, there is a maximal admissible graph (i.e., a facet) containing all edges of $G$, but not $e$. 
	
	\begin{corollary}[{\cite[Corollary 1.(2)]{GordonPetrov}}] 
		Let $\metric$ be a strict metric, then $f_0(\KRW{\metric}) = n \cdot (n+1)$. 
	\end{corollary}
	
	For $k \in \mathbb{N}$, denote by \[\CA{n}^k := \left\{H_{a,b} \left\vert
	\begin{array}{l}
		a,b\in [n]^k\\
		a_1,\ldots, a_k,b_1,\ldots, b_k \textrm{ mutually distinct}
	\end{array}
	\right.\right\}\]
	the set of hyperplanes in $\CA{n}$ corresponding to cycles of length $2k$ in $K_n$.
	\begin{definition}
		Let $\metric$ be a strict metric on $l$ points and define %the intersection 
		$X_\metric := \bigcap_{\substack{H \in \CA{l} \\ \metric \in H}} H $ and  $
		r_k(\metric) := \vert\{H \in \CA{l}^k \mid X_\metric\subseteq H\}\vert$. 
	\end{definition}
	
	\begin{lemma}\label{lem:2-adm}
		Let $\metric$ be a strict metric which is not contained in any hyperplane of $\CA{l}^2$. Then for each quadruple $x_1, y_1,x_2,y_2$ of vertices, if the edge set $\{(x_1,y_1),(x_2,y_2)\}$ 
		is admissible, it is face-defining. 
	\end{lemma}
%	\begin{proof}
%		Let $\{(x_1,y_1), (x_2,y_2)\}$ be a pair of admissible edges and let $e = (i,j)$ be an edge which is admissible together with the pair. We need to show that there is an admissible graph containing the pair, but not $e$, such that  the addition of $e$ would turn it inadmissible. We use the strictness of the metric and the strict inequality: \begin{equation} \label{ineq} \metric(x_1,y_1) + \metric(x_2,y_2) < \metric(x_1, y_2) + \metric(x_2, y_1). \end{equation}
%		
%		The trivial case is $\vert e \cap \{x_1,y_1,x_2,y_2\}\vert = 2$, then $e$ is either of $(x_1, y_2), (x_2, y_1)$. In this case, the graph we are looking for is the pair of edges $\{(x_1,y_1), (x_2,y_2)\}$ and the other edge, i.e., $(x_2, y_1)$ or $(x_1, y_2)$ since the set of edges $\{(x_1,y_1), (x_2,y_2),(x_1, y_2), (x_2, y_1)\}$ is inadmissible by assumption that $\metric$ is contained in no hyperplane in $\CA{l}^2$.
%		
%		For $\vert e \cap \{x_1,y_1,x_2,y_2\}\vert = 1$ or $0$: If $i$ is not contained in the quadruple, consider the edges $(x_1, i), (x_2, i)$, if $j$  is not contained in the quadruple, consider the edges $(j, y_1), (j,y_2)$. If we prove that one of these edges is admissible, we are finished: due to the strictness of $\metric$, the addition of $e$ would be inadmissible in either case, since $(i,j)$ together with any of these 4 edges would create a directed path. 
%		
%		We outline the argument for the edges $(j, y_1), (j,y_2)$. 
%		Assume to the contrary that both are inadmissible, meaning we have strict equations \[\metric(j, y_2) + \metric(x_2, y_1) < \metric(j, y_1) + \metric(x_2, y_2)\]\[\metric(j, y_1) + \metric(x_1, y_2) < \metric(j, y_2) + \metric(x_1, y_1).\] Adding the two equations, we get a contradiction to the inequality (\ref{ineq}).
%	\end{proof}
	With this, we can obtain the following general statement.
	
	
	\begin{proposition}
		Let $\metric$ be a strict metric on $l = n+1$ points. Then \[f_1(\KRW{\metric}) = \binom{n+2}{2, 2, n-2} -2\cdot r_2(\metric). \] 
	\end{proposition}
	
	Note that the multinomial coefficient is the number of faces in the generic case, where $X_\metric$ is the empty intersection. 
	
%	\begin{proof}
%		Let $\rho$ be a strict metric. By \Cref{strict_nopaths}, admissible  graphs on $2$ edges are of one of the two types depicted in \Cref{figure-ab1b2}.
%		\begin{figure}[ht]
%			\begin{subfigure}[t]{0.45\textwidth}
%				\centering
%				\begin{tikzpicture}
%					\node(x1) at (0,0) {$x_1$};
%					\node(x2) at (1,0) {$x_2$};
%					\node(y1) at (0,1) {$y_1$};
%					\node(y2) at (1,1) {$y_2$};
%					\draw[->] (x1.north) -- (y1.south);
%					\draw[->] (x2.north) -- (y2.south);
%				\end{tikzpicture}
%				\caption{Type (a)}
%			\end{subfigure}
%			\begin{subfigure}[t]{0.45\textwidth}
%				\centering
%				\begin{tikzpicture}
%					\node(x12) at (0,1) {$x_1=x_2$};
%					\node(y1) at (-1,0) {$y_1$};
%					\node(y2) at (1,0) {$y_2$};
%					\draw[->] (x12.south) -- (y1.north);
%					\draw[->] (x12.south) -- (y2.north);
%				\end{tikzpicture}
%				\begin{tikzpicture}
%					\node(x12) at (0,1) {$x_1=x_2$};
%					\node(y1) at (-1,0) {$y_1$};
%					\node(y2) at (1,0) {$y_2$};
%					\draw[<-] (x12.south) -- (y1.north);
%					\draw[<-] (x12.south) -- (y2.north);
%				\end{tikzpicture}
%				\caption{Type (b)}
%			\end{subfigure}
%			\caption{The two types of admissible graphs on two edges.}\label{figure-ab1b2}
%		\end{figure}
%		
%		
%		Each of the $\frac{(n+1)!}{(n-1)!}$ graphs of type (b) is always trivially admissible. Let us count the admissible graphs of type (a): for every choice of an ordered $4$-tuple $x_1,y_1,x_2,y_2$ consider the quantities
%		$$
%		s^+:= \rho(x_1,y_1) + \rho(x_2,y_2),\quad\quad
%		s^-:= \rho(x_1,y_2) + \rho(x_2,y_1).
%		$$
%		There are three cases: either $s^+<s^-$, in which case the graph with the edge set $\{(x_1,y_1),(x_2,y_2)\}$ is admissible and the one with the edge set $\{(x_1,y_2),(x_2,y_1)\}$ is not, or $s^+>s^-$, in which case the graph with the edge set $\{(x_1,y_2),(x_2,y_1)\}$ is admissible and the one with the edge set $\{(x_1,y_1),(x_2,y_2)\}$ is not, or $s^+=s^-$, in which case $\rho$ is contained in $H_{(x_1,x_2),(y_1,y_2)}\in \mathcal{W}_l^2$. 
%		
%		If $r_{2}(\rho)=0$ then the third case never enters and so exactly one of the other alternatives is realised. Moreover, in this setting the admissible graphs obtained in the first two cases are face-defining according to \Cref{lem:2-adm}, and we obtain a total of $\frac{(n+1)!}{4(n-3)!}$ admissible graphs of type (a). Thus, when $\rho$ is not contained in any member of $\mathcal{W}_l^2$ the proposition holds.
%		
%		
%		If $r_{2}(\rho)>0$, let $H_1,\ldots,H_{r_2(\rho)}$ be the hyperplanes of $\mathcal{W}_l^2$ containing $\rho$. The count of subgraphs of type (b) does not change. We now consider the graphs of type (a). Let $C_{H}$ be a cycle corresponding to $H \in \CA{l}^2$ (see Remark \ref{rem:cycle_arr}). What happens if we move from either halfspace onto the hyperplane $H$ is that all graphs remain admissible.
%		Moreover, if an admissible graph $G$ contains the set $C_{H}^+$ or $C_{H}^-$, respectively, the union of $G$ with $C_{H}^-$ or $C_{H}^+$, respectively, becomes admissible, too.
%		Thus, $C_{H}^+, C_{H}^-$ are no longer  faces of the polytope and we therefore subtract them from the count. Faces of dimension $1$ which correspond to paths on three vertices are not affected. 
%		
%		Without loss of generality, we can regroup the hyperplanes $H_1,\ldots,H_{r_2(\rho)}$ by the ground set of their underlying cycles.
%		Each quadruple $\{u,v,w,t\}$ either appears zero, once or three times, since the intersection \[H_{(v,w),(u,t)} \cap H_{(v,u),(w,t)}\cap H_{(v,t), (w,u)}\] has codimension $2$ for any quadruple $\{u,v,w,t\}$.
%		Thus if $\metric$ is in the intersection of two hyperplanes of cycles of one quadruple, it automatically is in the intersection of all three of them. 
%		
%		The first case is trivial, the case of the ground set appearing once is described above and in the last case, we subtract all $6$ faces ($2$ per cycle) from the generic count. 
%		Doing this iteratively for each quadruple gives the result.
%		
%	\end{proof}
	
%	For a low-dimensional example, see Table \ref{tab:metrics4}, where the $f$-vectors and corresponding inequalities are listed for $l = 4$. 
%	\begin{problem}
%		Considering the cases for vertices and edges, it is a natural question if the remaining entries of the $f$-vector can also be expressed by a formula using the binomial coefficients of the generic case and the integers $r_k(\metric), k \in \mathbb{N}$. 
%		
%		The higher cases are more complicated, because the admissibility condition for a graph involves its subgraphs as well, which means that there needs to be an inclusion exclusion argument. 
%	\end{problem}
	
	\section{Explicit computations}\label{sec:compuations}
	
	In this section, we explicitly study the KRW polytopes of metrics on $4$, $5$, and $6$ points.
	We published an accompanying data set of one generic metric of each combinatorial type of these sizes on Zenodo~\cite{zenodo}.
	
	\subsection{The case \texorpdfstring{$n=4$}{n=4}.}\label{sec_n4}
	There is only one generic KRW polytope up to symmetry. % The Wasserstein arrangement is depicted in \Cref{fig:arrangement4} where the cycles corresponding to the hyperplanes are shown next to them. 
	
	In this case, the arrangement agrees with the braid arrangement $\mathcal{A}_3$ up to the lineality space and thus has 6 chambers. 
	
	There are three different types of strict, not generic metrics, two of which have the same $f$-vector, but different combinatorial type. 
	%The strict but not generic cases are illustrated in items (2-4) in  Table \ref{tab:metrics4}. Items 2 and 3 are metrics on the rank one intersections of the hyperplane arrangement, and item 4 is on a rank 2 intersection. Observe that while t
	The $f$-vector only depends on the rank of the intersection of the corresponding hyperplanes while their tight spans differ. 
%	\begin{figure}
%		\centering 
%		\includegraphics[width=0.8\textwidth]{figures/Zeichnung_arrangement_4} 
%		\caption{Wasserstein arrangement for $n = 4$. Each hyperplane corresponds to a cycle as described in Remark \ref{rem:cycle_arr} and each chamber has one set of oriented edges of each cycle that is admissible. We  obtain the admissible graphs of a neighboring chamber by applying the permutation specified on the hyperplane separating the chambers. The edge pairs in the grey area are the ones corresponding to the horizontal hyperplane. } 
%		\label{fig:arrangement4}
%	\end{figure}
%	\begin{small} 
%	\newcolumntype{C}{>{\centering\arraybackslash}m{7em}}
%	\renewcommand\arraystretch{0.5}
%	\begin{table}\sffamily
%		\begin{tabular}{l*5{C}@{}}
%			\toprule
%			\textsc{Nr.} & \textsc{case} & \textsc{sample metric} & \textsc{$f$-vector} & \textsc{polytope} & \textsc{tight span}  \\ 
%			\midrule
%			1 & $d_{12|34} > d_{13|24} > d_{14|23}$ & $\begin{pmatrix}
%				0&8&7&5 \\ 8&0 &5&7\\7&5&0&8\\ 5 & 7 &8 &0
%			\end{pmatrix}$ & (12 30 20) & \includegraphics[width=6em]{figures/generic_example.png} & \includegraphics[width=6em]{figures/Zeichnung_TS_generic.pdf}\\ 
%			2 &  $d_{12|34} > d_{13|24} = d_{14|23}$& $\begin{pmatrix}
%				0&6&5&5 \\ 6&0 &5&5\\5&5&0&6\\ 5 & 5 &6 &0
%			\end{pmatrix}$ & (12 28 18) & \includegraphics[width=6em]{figures/example_2.png} & \includegraphics[width=4em]{figures/Zeichnung_TS_almost_generic.pdf} \\ 
%			3 & $d_{12|34} = d_{13|24} > d_{14|23}$ & $\begin{pmatrix}
%				0&3&3&2 \\ 3&0 &2&3\\3&2&0&3\\ 2 & 3 &3 &0
%			\end{pmatrix}$ & (12 28 18) & \includegraphics[width=6em]{figures/example_3.png} & \includegraphics[width=6em]{figures/Zeichnung_TS_fast_generic.pdf} \\ 
%			4 & $d_{12|34} = d_{13|24} = d_{14|23}$ & $\begin{pmatrix}
%				0&1&1&1 \\ 1&0 &1&1\\1&1&0&1\\ 1 & 1 &1 &0
%			\end{pmatrix}$ & (12 24 14) & \includegraphics[width=6em]{figures/root_polytope.png} & \includegraphics[width=3em]{figures/Zeichnung_TS_rootpolytope.pdf} \\ 
%			\bottomrule 
%		\end{tabular}
%		\caption{Types of strict metrics for $n = 4$, their KRW polytopes and the corresponding tight spans.}
%		\label{tab:metrics4}
%	\end{table} 
%	\end{small} 
%	\begin{figure}
%		\centering 
%		\includegraphics[width=0.5\textwidth]{figures/Zeichnung_tight_span_4.pdf} 
%		\caption{Wasserstein arrangement for $n= 4$, coded by the tight spans. For detailed encoding of the arrangement see Figure  \ref{fig:arrangement4} and the numbers 1-4 correspond to the tight spans in Table \ref{tab:metrics4}. Number 4 is the cone of convex combinations of elementary splits.} 
%		\label{fig:tight_span_4}
%	\end{figure}
%	
	
	\subsection{The case \texorpdfstring{$n=5$}{n=5}.}\label{n_gleich_5}
	The characteristic polynomial of the Wasserstein arrangement~$\CA{5}$ is $\chi_{\CA{5}}(t)=t^{10} - 15t^9 + 90t^8 - 260t^7 + 350t^6 - 166t^5$ which does not factor over the integers.
	Thus by Zaslavsky's theorem, the arrangement~$\CA{5}$ has $882$ chambers which agrees with the number of maximal cones of the fan $\GPF{5}$~\cite{Zas75}.
	There are 12 generic KRW polytopes up to symmetry with $f$-vector (20 90 140 70) that are pairwise not isomorphic. %The $f$-vector of these polytopes is (20 90 140 70), as was proved in \cite{GordonPetrov}. 
	
	Moreover, there are 65 combinatorial types of strict but not generic KRW polytopes. 
	We computed these using the \texttt{polymake}~\cite{polymake} hyperplane arrangement package~\cite{polymake_arrangements} by enumerating all cells of the Wasserstein arrangement.
	%We record descriptions of these 65 combinatorial types and their metrics in~\Cref{sec:strict}.
	
	% \subsection{The table}
%	Now, we proceed with a description of the $12$ combinatorial types of generic KRW polytopes.
%	The encoding of the chambers is to be understood as follows: The even cycles on 5 points are just the 4-cycles. Each 4-cycle can be encoded with the notation as seen in \Cref{fig:arrangement4} by the cyclic permutations of length two. Consider the following matrix. The $i$-row of this matrix contains the cycles on the ground set $[5]\backslash \{i\}$. For example $M_{12}$ encodes the cycle $[2,3,5,4]$, whereas $M_{22}$ encodes the cycle $[1,3,5,4]$.  
%	\[M = \begin{bmatrix} (3 5) & (4 5) & (3 4) \\ (3 5) & (4 5) & (3 4) \\ (2 5) & (4 5) & (2 4) \\ (2 5) & (3 5) & (2 3) \\ (2 4) & (3 4) & (2 3)  \end{bmatrix}\] 
%	
%	
%	Again using \Cref{fig:arrangement4}, we define the side of the horizontal pair of edges to be positive and can thus encode the structure of the chamber in a $(5\times 3)$-matrix with 1, -1 entries according to the sign of the chamber regarding the hyperplane specified by the above encoding. 
	
	The symmetric group $S_5$ acts on the chambers by coordinate permutation. %The number of chambers yielding the same metric as a given chamber is 120 divided by the order of the stabilizer of the chamber under this action. 
	
	Two chambers yield combinatorially equivalent polytopes if and only if there is $\sigma \in S_5$ mapping one chamber to the other. 
%	
%	\begin{small}
%		\renewcommand\arraystretch{0.5}
%		\begin{longtable}{@{}lllll p{1in}@{}}
%			\toprule
%			
%			\textsc{Nr.} & \textsc{Chambers} & \textsc{sample metric} & \textsc{Schlegel diagram} & \textsc{stabilizer} \\
%			
%			\midrule
%			1 & $\begin{bmatrix} 1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1  \end{bmatrix}$ & $\begin{pmatrix}
%				0&16&13&11&9 \\ 16&0 &9&9& 11\\13&9&0&9 &13\\ 11& 9 &9 &0 & 16 \\ 9 & 11 & 13 & 16 & 0 
%			\end{pmatrix}$ & \raisebox{-0.5\totalheight}{\includegraphics[width =12em]{figures/Metrik_1_zuschnitt.png}} & $\langle (1 5)(2 4)\rangle$ \\
%			2 & $\begin{bmatrix} 1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & -1 & 1  \end{bmatrix}$& $\begin{pmatrix}
%				0&14&14&11& 8 \\ 14&0 &8&9 & 10\\14&8&0&7 & 12\\ 11 & 9&7 &0 &15 \\ 8 & 10 & 12 & 15 & 0 
%			\end{pmatrix}$ &  \raisebox{-0.5\totalheight}{\includegraphics[width =12em]{figures/Metrik_2_zuschnitt.png}} & \groupnamea \\ 
%			3 & $\begin{bmatrix} 1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & -1  \end{bmatrix}$& $\begin{pmatrix}
%				0&48&35&34 & 26 \\ 48&0 &31&26&34\\35&31&0&26 & 40\\ 34 & 26 &26 &0 & 48 \\ 26 & 34 & 40& 48 & 0
%			\end{pmatrix}$ & \raisebox{-0.5\totalheight}{\includegraphics[width =12em]{figures/Metrik_3_zuschnitt.png}} & \groupnamea \\
%			4 & $\begin{bmatrix} 1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & -1 & -1  \end{bmatrix}$& $\begin{pmatrix}
%				0&12&14&12 & 7 \\ 12&0 &10&10 & 9\\14&10&0&6 & 13\\ 12 & 10 & 6&0 & 15 \\ 7 & 9 & 13 & 15 & 0 
%			\end{pmatrix}$ &  \raisebox{-0.5\totalheight}{\includegraphics[width =12em]{figures/Metrik_4_zuschnitt.png}} & \groupnamea \\
%			5 & $\begin{bmatrix} 1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\-1 & -1 & -1  \end{bmatrix}$ & $\begin{pmatrix}
%				0&12&13&14&6 \\ 12&0 &13&10 & 10\\13&13&0&7 & 13\\ 14 & 10 &17 &0 & 16 \\ 6 & 10 & 13 & 16 & 0
%			\end{pmatrix}$ &  \raisebox{-0.5\totalheight}{\includegraphics[width =12em]{figures/Metrik_5_zuschnitt.png}} & $\langle (1 3) (4 5) \rangle$\\
%			6 & $\begin{bmatrix} 1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\-1 & 1 & -1  \end{bmatrix}$ & $\begin{pmatrix}
%				0&42&35&40&20 \\ 42&0 &37&26&34\\35&37&0&23&37\\ 40 & 26 &23 &0 & 48 \\ 20 & 34 & 37 & 48 & 0 
%			\end{pmatrix}$ & \raisebox{-0.5\totalheight}{\includegraphics[width =12em]{figures/Metrik_6_zuschnitt.png}} & \groupnamea \\ 
%			7 & $\begin{bmatrix} 1 & 1 & -1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & -1  \end{bmatrix}$ & $\begin{pmatrix}
%				0&54&36&41&25 \\ 54&0 &36&25&41\\36&36&0&36 & 36\\ 41 & 25 &36 &0 & 54 \\ 25 & 41 & 36 & 54 & 0 
%			\end{pmatrix}$ & \raisebox{-0.5\totalheight}{\includegraphics[width =12em]{figures/Metrik_7_zuschnitt.png}} & $\langle (1 2)(4 5), (1 4)(2 5)\rangle$\\ 
%			8 & $\begin{bmatrix} 1 & -1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & -1 & 1  \end{bmatrix}$  & $\begin{pmatrix}
%				0&48&52&38&23 \\ 48&0 &26&40 & 37\\52&26&0&26 & 51\\ 38 & 40 &26 &0 & 49 \\ 23 & 37 & 51 & 49 & 0 
%			\end{pmatrix}$ & \raisebox{-0.5\totalheight}{\includegraphics[width =12em]{figures/Metrik_8_zuschnitt.png}} & $\langle (1 5)(2 4) \rangle$\\ 
%			9 & $\begin{bmatrix} 1 & -1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\-1 & -1 & 1  \end{bmatrix}$  & $\begin{pmatrix}
%				0&12&16&13& 6 \\ 12&0 &10&13& 10\\16&10&0&7 & 16\\ 13 & 13 &7 &0 & 15 \\ 6 & 10 & 16 & 15 & 0 
%			\end{pmatrix}$ & \raisebox{-0.5\totalheight}{\includegraphics[width =12em]{figures/Metrik_9_zuschnitt.png}} & $\langle (1 4)(3 5) \rangle$\\ 
%			10 & $\begin{bmatrix} 1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & -1 \\1 & 1 & 1 \\1 & 1 & 1  \end{bmatrix}$ & $\begin{pmatrix}
%				0&48&37&29 & 31 \\ 48&0 &29&31& 29\\37&29&0&29 & 37 \\ 29 & 31 &29 &0 & 48 \\ 31 & 29 & 37 & 48 & 0 
%			\end{pmatrix}$ & \raisebox{-0.5\totalheight}{\includegraphics[width =12em]{figures/Metrik_10_zuschnitt.png}} & $\langle (1 5)(2 4) \rangle$\\ 
%			11 & $\begin{bmatrix} 1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & -1 \\1 & 1 & 1 \\1 & -1 & 1  \end{bmatrix}$ & $\begin{pmatrix}
%				0&48&52&35& 34 \\ 48&0 &32&43& 26\\52&32&0&29&40\\ 35 & 43 &29 &0 & 57 \\ 34 & 26 & 40 & 57 & 0
%			\end{pmatrix}$ & \raisebox{-0.5\totalheight}{\includegraphics[width =12em]{figures/Metrik_11_zuschnitt.png}} &  $\langle (2 3)(4 5) \rangle$\\ 
%			12 & $\begin{bmatrix} 1 & -1 & 1 \\1 & 1 & 1 \\1 & 1 & -1 \\1 & 1 & 1 \\1 & -1 & 1  \end{bmatrix}$ & $\begin{pmatrix}
%				0&36&40&23 & 21 \\ 36&0 &23&40 & 21\\40&23&0&23& 38\\ 23 & 40 &23 &0 & 38 \\ 21 & 21 & 38 & 38 & 0
%			\end{pmatrix}$ &  \raisebox{-0.5\totalheight}{\includegraphics[width =12em]{figures/Metrik_12_zuschnitt.png}} & $D_{10}$ in $S_5$ (\footnote{We write $D_{10}$ for the dihedral group with $10$ elements, see Table~\ref{tab:subgroups_s5}.})\\ 
%			\bottomrule
%			\caption{Types of generic metrics for $n = 5$.}
%			\label{tab:metrics5_gen}
%		\end{longtable} 
%	\end{small}
	
	
	\subsection{The case \texorpdfstring{$n=6$}{n=6}.}
	For $6$-point metrics, $\CA{6}$ is an arrangement in $\R^{15}$ with a $6$-dimensional lineality space consisting of $105$ hyperplanes, see \Cref{prop:size_wn}.
	%We found that this arrangement has $6,677,863,200$ chambers (this agrees by the above discussion with the number of combinatorial types of generic KRW polytopes).
	
	The generic KRW polytopes correspond exactly  to certain regular triangulations of $RP^0_n$, the (full) root polytope together with the origin.
	Jörg Rambau computed all symmetric regular triangulations of $RP^0_6$ up to symmetry using the latest version of his software \texttt{TOPCOM}~\cite{
		Ram23}.
	There are exactly $25,224$ such triangulations.
	%This number therefore agrees with the number of combinatorial types of KRW polytopes of $6$-point metrics up to the action of the symmetry group $S_6$.
	%We record both statements in the following theorem.
	
	\begin{theorem}
		There are $25,224$ generic KRW polytopes up to symmetry (action of $S_6$) that are pairwise not isomorphic.
	\end{theorem}
	
	
%	
%	
%	
%	
%	
%	
%	
%	\renewcommand*{\bibfont}{\footnotesize}
%	\printbibliography
%	\newpage
%	\appendix
%	\section{Table of strict KRW polytopes on $5$ points}\label{sec:strict}
%	
%	In the following table, each row represents one of the $65$ combinatorial types of KRW polytopes of strict metric spaces on $5$ points. For each combinatorial type, we list a sample metric $d$, the $f$-vector of the polytope $\KRW{d}$ and the stabilizer subgroup of the cone of $\GPF{5}$ containing $d$ (see \Cref{n_gleich_5}).
%	
%	The sample metrics are encoded as vector tuples $(d_{11},d_{12}, d_{13}, d_{14}, d_{15}, d_{23}, d_{24}, d_{25}, d_{34}, d_{35}, d_{45})$. See \Cref{tab:subgroups_s5} below for the encoding of the stabilizer subgroups.
%	
%	
%	\renewcommand\arraystretch{0.5}
%%	
%%	\begin{longtable}{@{}llll p{1in}@{}}
%%		\toprule
%%		\textsc{$f$-vector} &\textsc{Nr.} & \textsc{sample metric} & \textsc{stabilizer}\\
%%		
%%		\midrule
%%		20 60 70 30 &1.1 & 2 2 2 2 2 2 2 2 2 2&$S_5$  \\
%%		&&& \\ 
%%		20 72 94 42 &2.1 & 6 5 5 5 5 5 5 6 6 6&$\groupnameg$  \\ 
%%		& 2.2 & 
%%		3 3 3 2 2 2 3 2 3 3&$\groupnameg$  
%%		
%%		\\ 
%%		
%%		&&& \\ 20 76 102 46 & 3.1 & 
%%		6 5 5 5 5 5 5 4 6 6&$\groupnamec$
%%		\\ &&& 
%%		\\ 20 80 112 52 & 4.1 &
%%		8 7 7 5 5 5 7 6 8 8&$\groupnameb$ 
%%		
%%		\\ & 4.2 & 
%%		8 7 6 6 7 6 6 5 7 8&$\groupnameb$
%%		
%%		\\ & 4.3 & 
%%		8 7 7 5 5 5 7 4 8 8 &$\groupnameb$
%%		\\ 
%%		
%%		&&& \\ 20 80 114 54 & 
%%		5.1 & 10 8 7 7 8 7 7 8 8 10&$\groupnamee$ 
%%		
%%		\\ 
%%		& 5.2 & 3 4 4 2 3 3 3 2 4 4&$\groupnamee$  
%%		
%%		\\ & 5.3&
%%		4 3 3 3 3 3 3 3 4 4&$\groupnamec$ 
%%		\\ 
%%		& 5.4 & 
%%		6 7 7 5 7 7 5 4 8 8&$\groupnamec$
%%		
%%		\\ 
%%		
%%		&&& \\ 20 82 116 54 & 6.1 & 
%%		6 7 6 5 5 6 5 5 6 7& $\groupnameb$
%%		
%%		\\ &&&
%%		\\ 20 84 122 58&  7.1 & 
%%		8 7 6 6 5 6 6 5 7 8&$\groupnameb$
%%		
%%		\\ & 7.2 & 
%%		8 8 7 6 6 7 6 5 8 9&\groupnamea 
%%		
%%		\\ & 7.3 & 
%%		8 9 7 6 5 7 6 6 7 9& \groupnamea 
%%		
%%		\\ &  7.4 & 
%%		8 9 7 6 7 7 6 6 7 9&\groupnamea 
%%		\\ & 7.5 & 
%%		8 9 8 5 5 6 7 5 8 9 &\groupnamea  \\ & 7.6 & 
%%		
%%		8 7 9 5 9 7 7 6 8 10&$\groupnameb$
%%		\\ & 7.7 & 
%%		10 9 8 6 7 6 8 5 9 10&\groupnamea \\ 
%%		&&& \\ 20 84 124 60 &  8.1 & 
%%		12 9 8 8 9 8 8 8 10 12 &$\groupnameb$
%%		\\ & 8.2  & 
%%		5 4 4 3 3 3 4 3 5 5&$\groupnameb$ \\ &  8.3 & 
%%		8 9 8 6 9 8 6 5 9 10&$\groupnameb$ 
%%		\\ & 8.4 & 
%%		8 9 9 5 7 7 7 4 10 10&$\groupnameb$
%%		\\ &&& \\
%%		20 86 128 62 & 9.1& 
%%		12 10 9 7 8 7 9 8 10 12&$\groupnameb$ 
%%		\\
%%		& 9.2 & 
%%		
%%		10 8 7 7 6 7 7 7 9 10 &\groupnamea \\ & 9.3& 
%%		10 9 8 6 5 6 8 5 9 10& $\groupnameb$ 
%%		\\ & 9.4 & 
%%		10 10 8 7 6 8 7 6 9 11&\groupnamea 
%%		\\ & 9.5 & 
%%		10 10 8 7 8 8 7 6 9 11&\groupnamea \\ 
%%		& 9.6 & 
%%		10 11 8 7 7 8 7 7 8 11 & \groupnamea  \\ & 9.7 & 
%%		10 10 9 6 6 7 8 5 10 11&\groupnamea 
%%		\\ & 9.8 & 
%%		10 11 9 6 5 7 8 6 9 11 & \groupnamea   \\ 
%%		& 9.9 & 
%%		10 11 9 6 7 7 8 6 9 11&\groupnamea  
%%		\\ & 9.10 & 
%%		30 26 25 17 22 17 25 17 25 30&\groupnamea 
%%		\\ & 9.11 & 
%%		36 26 25 23 28 23 25 26 28 36&$\groupnameb$  \\ & 9.12 & 
%%		
%%		30 23 22 20 25 20 22 17 25 30&\groupnamea
%%		\\  & 9.13 & 
%%		8 10 9 5 8 7 7 5 9 10&$\groupnameb$
%%		\\ & 9.14 & 
%%		8 10 9 6 8 9 6 5 10 11&\groupnamea 
%%		\\ & 9.15 & 
%%		
%%		8 11 10 5 7 8 7 5 10 11&$\groupnameb$ 
%%		\\ & 9.16& 
%%		10 9 10 6 11 8 8 7 9 12&$\groupnameb$ 
%%		\\ & 9.17 & 
%%		10 9 11 5 9 7 9 6 10 12&\groupnamea
%%		\\ & 9.18 & 
%%		30 23 31 17 31 23 25 20 28 36&\groupnamea
%%		\\ & 9.19 & 
%%		10 12 9 6 8 11 8 7 12 11&$\groupnameb$
%%		\\ & 9.20 & 
%%		
%%		30 25 20 22 17 22 20 17 25 30&$\groupnameb$
%%		\\ &&& \\ 20 88 134 66& 10.1 & 
%%		14 11 10 8 9 8 10 8 12 14& \groupnamea 
%%		\\ & 10.2 & 
%%		14 11 9 9 9 9 9 9 11 14&$\groupnameb$
%%		\\ & 10.3 & 
%%		12 10 9 7 6 7 9 7 11 12&\groupnamea
%%		\\& 10.4 & 
%%		12 12 9 8 8 9 8 7 10 13&\groupnamea
%%		\\ & 10.5 & 
%%		12 12 10 7 6 8 9 6 11 13&\groupnamea
%%		\\ & 10.6 & 
%%		12 12 10 7 8 8 9 6 11 13& \groupnamea
%%		\\ & 10.7 & 
%%		12 13 10 7 7 8 9 7 10 13&\groupnamea
%%		\\& 10.8 & 
%%		42 32 31 23 28 23 31 26 34 42&\groupnamea
%%		\\ & 10.9 & 
%%		36 29 28 20 25 20 28 17 31 36&\groupnamea
%%		\\ & 10.10 & 
%%		
%%		42 29 28 26 31 26 28 26 34 42&\groupnamea
%%		\\ & 10.11 & 
%%		10 11 10 6 9 8 8 5 11 12&\groupnamea
%%		\\ & 10.12 & 
%%		10 12 10 7 10 10 7 6 11 13&\groupnamea
%%		\\ & 10.13& 
%%		10 12 11 6 8 9 8 5 12 13&\groupnamea
%%		\\ & 10.14 & 
%%		10 13 11 6 9 9 8 6 11 13&\groupnamea 
%%		\\ & 10.15 &
%%		12 11 12 6 11 8 10 7 11 14&\groupnamea
%%		\\ & 10.16 & 
%%		10 11 13 5 11 9 9 6 12 14&$\groupnameb$
%%		\\ & 10.17 & 
%%		36 29 34 20 37 26 28 23 31 42&\groupnamea
%%		\\ & 10.18 & 
%%		36 29 37 17 31 23 31 20 34 42&\groupnamea
%%		\\ & 10.19 & 
%%		42 46 32 23 26 40 31 26 45 43&$\groupnameb$ 
%%		\\ & 10.20 & 
%%		12 14 11 6 8 11 10 7 14 13&\groupnamea
%%		\\ & 10.21 & 
%%		36 28 23 25 20 25 23 23 31 36&\groupnamea
%%		\\ & 10.22 & 
%%		42 46 32 31 26 40 23 26 37 51 &$\groupnameb$  \\ 
%%		
%%		\bottomrule
%%		\caption{Types of strict metrics for $n = 5$.}
%%		\label{tab:metrics5_strict}
%%	\end{longtable} 
%%	
%%	
%%	\newcolumntype{D}{>{\centering\arraybackslash}m{6em}}
%%	\begin{table}[h]\sffamily
%%		\begin{tabular}{l*4{D}@{}}
%%			\toprule
%%			\textsc{Description: Subgroup of $S_5\dots$} & \textsc{order} & \textsc{example} & \textsc{name}  \\ 
%%			\midrule
%%			trivial & 1&  $()$ & \groupnamea \\ 
%%			gen. by double transposition %in $S_5$ 
%%			& 2& $\langle(1 5)(2 4)\rangle$ &  $\groupnameb$ \\ gen. by pair of disjoint transpositions %in $S_5$
%%			& 4& $\langle (1 2)\comma (3 4) \rangle$ &  $\groupnamec$ \\ 
%%			gen. by double transpositions &&&\\ 
%%			\hspace{1em} on 4 elements %in $S_5$ 
%%			& 4& $\langle(1 2)(4 5)\comma(1 4)(2 5)\rangle$& $\groupnamed$ \\ 
%%			
%%			dihedral of order $8$ %subgroup of $S_5$ 
%%			& 8 & $\langle (1 2 3 4)\comma (1 3)\rangle$ & $\groupnamee$ \\ 
%%			
%%			dihedral of order $10$
%%			
%%			& 10 & $\langle (1 2 3 4 5)\comma (2 5)(3 4)\rangle$ & $\groupnamef$ \\ 
%%			direct product of $S_3$ and $S_2$ 
%%			
%%			& 12 & $\langle (1 2 3)\comma (1 2)\comma (4 5) \rangle$ & $\groupnameg$ \\ 
%%			\bottomrule 
%%		\end{tabular}
%%		\caption{List of types of subgroups of $S_5$ appearing as stabilizer of chambers in $\W_5$.}
%%		\label{tab:subgroups_s5}
%%	\end{table} 
%	
\printbibliography
\end{document}


%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way





