%% 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]{FPSAC2023}
\articlenumber{48}
%% 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}

%%%%%%%%%% OUR COMMANDS

\usepackage{mathrsfs}
\usepackage{textcomp}
\usepackage{tikz}

\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}[lemma]{Proposition}
\newtheorem{theorem}[lemma]{Theorem}
\newtheorem{corollary}[lemma]{Corollary}
\newtheorem{conjecture}[lemma]{Conjecture}
\newtheorem{example}[lemma]{Example}
\theoremstyle{definition}
\newtheorem{definition}[lemma]{Definition}
\newtheorem{remark}[lemma]{Remark}

\newcommand{\Bbbb}{\mathbb{B}}
\newcommand{\Ball}{\mathbb{B}}
\newcommand{\Ccc}{\mathbb{C}}
\newcommand{\Fff}{\mathbb{F}}
\newcommand{\Nnn}{\mathbb{N}}
\newcommand{\Rrr}{\mathbb{R}}
\newcommand{\Sss}{\mathbb{S}}
\newcommand{\Qqq}{\mathbb{Q}}
\newcommand{\Zzz}{\mathbb{Z}}
\newcommand{\R}{\Rrr}
\newcommand{\Q}{\Qqq}
\newcommand{\Z}{\Zzz}
\newcommand{\bl}[2]{(#1,#2)}
\newcommand{\circdots}{\circ \cdots \circ}
\newcommand{\F}{\mathcal{F}}
\newcommand{\Bf}{\mathcal{B}}
\newcommand{\Cf}{\mathcal{C}}
\newcommand{\Hf}{\mathcal{H}}
\newcommand{\Tc}{\mathscr{T}}
\newcommand{\Tt}{\mathcal{T}}
\newcommand{\Zf}{\mathcal{Z}}
\newcommand{\Pf}{\mathcal{P}}
\newcommand{\Zfe}{\mathcal{Z_{\mathrm{ext}}}}
\newcommand{\Zfep}{\mathcal{Z_{\mathrm{ext}}'}}
\newcommand{\Pfe}{\mathcal{P_{\mathrm{ext}}}}
\newcommand{\fl}{\longrightarrow}
\DeclareMathOperator{\im}{imult}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\Vol}{Vol}
\DeclareMathOperator{\Ker}{Ker}

%%%%%%%%%% END OF OUR COMMANDS


%% define your title in the usual way
\title[Pizza and $2$-structures]{Pizza and $2$-structures}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[Richard Ehrenborg, Sophie Morel, Margaret Readdy]
{Richard Ehrenborg\thanks{\href{richard.ehrenborg@uky.edu}{richard.ehrenborg@uky.edu}.
Partially supported by the Simons Foundation
(\#429370 to Richard~Ehrenborg).}\addressmark{1},
Sophie Morel\thanks{\href{sophie.morel@ens-lyon.fr}{sophie.morel@ens-lyon.fr}. 
Partially 
supported by the LABEX MILYON (ANR-10-LABX-0070) of Universit\'e 
de Lyon, within the program ``Investissements d'Avenir'' (ANR-11-IDEX-0007)
operated by the French National Research Agency (ANR),
and by Princeton University.}\addressmark{2},
\and
Margaret Readdy\thanks{\href{margaret.readdy@uky.edu}{margaret.readdy@uky.edu}.
Partially supported by the Simons Foundation
(\#422467 to Margaret Readdy).}\addressmark{1}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}
Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA \\ 
\addressmark{2}
ENS de Lyon, Unit\'e De Math\'ematiques Pures Et Appliqu\'ees, 69342 Lyon Cedex 07, France \\
%\addressmark{3}
%Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA
}

%% 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{Let $\mathcal{H}$ be a Coxeter hyperplane arrangement 
in $n$-dimensional Euclidean space.
Assume that the negative of the identity map belongs to the associated Coxeter group~$W$.
Furthermore assume that the arrangement is not of type $A_1^n$.
Let $K$ be a measurable subset of the Euclidean space with finite volume
which is stable by the Coxeter group~$W$ and let $a$ be a point 
such that $K$ contains the convex hull of the orbit of the point~$a$ under the group~$W$.
In a previous article the authors proved the
generalized pizza theorem:
that the alternating sum over the chambers~$T$ of~$\mathcal{H}$ of
the volumes of the intersections $T\cap(K+a)$ is zero.
In this paper we give a dissection proof of this result.
In fact, we lift the identity to an abstract dissection
group to obtain a similar identity that replaces the volume by any
valuation that is invariant under affine isometries.
This includes the cases of all intrinsic volumes. 
Apart from basic geometry, the main ingredient is 
a previous theorem of the authors 
where we relate the alternating sum
of the values of certain valuations over the chambers of a Coxeter arrangement
to similar alternating sums for simpler subarrangements called
$2$-structures,
introduced by Herb to study discrete series characters of real
reduced groups.
}

%% put your French abstract here, or comment this out if you don't have one
\resume{Soit $\mathcal{H}$ un arrangement d'hyperplans de Coxeter dans un espace
euclidien $V$ de dimension $n$. On suppose que $\mathcal{H}$ n'est pas
de type $A_1^n$ et que l'isom{\'e}trie $x\mapsto -x$ sur $V$ appartient {\`a}
son groupe de Coxeter~$W$. Soit $K$ un sous-ensemble
mesurable de mesure finie de $V$ qui est stable par $W$, et soit $a$ un
point de $V$ tel que $K$ contienne l'enveloppe convexe de l'orbite de
$a$ sous~$W$. Dans un article pr{\'e}c{\'e}dent, nous avons prouv{\'e} le ``th{\'e}or{\`e}me
de la pizza g{\'e}n{\'e}ralis{\'e}'', qui dit que la somme altern{\'e}e sur les chambres
$T$ de $\mathcal{H}$ du volume de $T\cap(K+a)$ est nulle. Dans cet
article, nous donnons une preuve g{\'e}om{\'e}trique par dissection de ce
r{\'e}sultat. Plus pr{\'e}cis{\'e}ment, nous relevons l'identit{\'e} ci-dessus dans un
groupe de dissection abstrait, ce qui implique des identit{\'e}s similaires
lorsque le volume est remplac{\'e} par n'importe quelle valuation (mesure
finiment additive) invariante par isom{\'e}tries affines. Ceci inclut en
particulier le cas des volumes intrins{\`e}ques sur les ensembles convexes
compacts. Les ingr{\'e}dients principaux de la preuve sont de la g{\'e}om{\'e}trie
{\'e}l{\'e}mentaire et un th{\'e}or{\`e}me pr{\'e}c{\'e}dent des auteurs qui calcule la somme
altern{\'e}e des valeurs d'une valuation sur les chambres d'un arrangement
de Coxeter en fonction de sommes altern{\'e}es similaires associ{\'e}es {\`a} des
sous-arrangements plus simples appel{\'e}s $2$-structures,
introduits {\`a} l'origine par Herb pour {\'e}tudier les caract{\`e}res des s{\'e}ries
discr{\`e}tes des groupes r{\'e}ductifs r{\'e}els.
}


%% put your keywords here, or comment this out if you don't have them yet
\keywords{Coxeter arrangements,
$2$-structures,
dissections,
Pizza Theorem,
reflection groups,
intrinsic volumes,
pseudo-root systems,
Bolyai-Gerwien theorem}

%% 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]{biblatex}
\addbibresource{2-structures_FPSAC.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}

The $2$-dimensional pizza theorem is the following result: Given a disc in the plane,
choose a point on this disc and cut
the disc by $2k$ equally spaced lines passing through the point,
where $k \geq 2$. 
The alternating sum of the areas of the resulting slices is then equal to zero.
This was first proved by Goldberg~\cite{Gol}.  Frederickson~\cite{Fred}
gave a dissection proof based on dissection proofs
of Carter-Wagon in the case $k=2$ (see~\cite{Carter_Wagon}) and
of Allen Schwenk (unpublished) in the cases $k=3,4$. 
Frederickson deduced
dissection proofs of a similar sharing result for the pizza crust and
of the so-called calzone theorem, which is the analogue of the
pizza theorem for a ball in $\R^3$ that is cut by one horizontal
plane and by $2k$ equally-spaced vertical planes all meeting
at one point in the ball.

To generalize the pizza problem, consider a finite central hyperplane
arrangement~$\Hf$ in~$\R^n$ and fix a base chamber of this arrangement.
Each chamber~$T$ has a sign $(-1)^{T}$ determined by the parity of the number of
hyperplanes separating it from the base chamber. If $K$ is a measurable
subset of~$\R^n$ of finite volume, what can we say about
the pizza quantity $\sum_T (-1)^T\Vol(T\cap K)$, 
where the sum runs over all the chambers $T$ of~$\Hf$?
The original pizza theorem is the case where $n=2$, $\Hf$ has the type of
the dihedral arrangement $I_2(2k)$ 
and $K$ is a disc containing the origin.
The calzone theorem is the case where $n=3$, $\Hf$ has the type of the product
arrangement $I_2(2k)\times A_1$ and $K$ is a ball containing the origin.





The following generalization of the pizza and the calzone was
proved in~\cite[Corollary~6.9]{EMR_pizza} by analytic means.
We also recently learned that Brailov had proved independently
the same theorem for a ball and an arrangement of type $B_n$ in~\cite{Brailov},
by somewhat similar methods.

\begin{theorem}[Ehrenborg, Morel and Readdy]
Let $\Hf$ be a Coxeter arrangement with Coxeter group~$W$
that contains the negative of the identity map, denoted by $-\id$.
Assume that $\Hf$ is not of type $A_{1}^{n}$.
Let $K$ be a set of finite measure that is 
stable by the group~$W$.
Then for every point $a\in\R^n$ such that $K$ contains the convex hull of $\{w(a) : w \in W\}$, 
we have
\[\sum_T (-1)^T\Vol(T\cap(K+a))=0 .\]
\label{theorem_introduction_first}
\end{theorem}
The proof of this result uses an expression for $\sum_T(-1)^T\Vol(T\cap (K+a))$ 
as an alternating sum of pizza quantities
over subarrangements of $\Hf$ of the form
$\{e_1^\perp,\ldots,e_n^\perp\}$ with $(e_1,\ldots,e_n)$ an
orthonormal basis of $\R^n$, in other words, subarrangements
that have type~$A_1^n$.



In the paper~\cite{EMR}, we study a different sum
$\sum_T (-1)^T\nu(\overline{T})$, where $\nu$ is a valuation
defined on closed convex polyhedral cones of $\R^n$ that takes 
integer values.
Under the same condition that $\Hf$ is a Coxeter arrangement,
we rewrite this quantity as an alternating
sum of similar quantities for certain subarrangements of $\Hf$ that are products
of rank~$1$ and rank~$2$ arrangements~\cite[Theorem~3.2.1]{EMR},
and then deduce an expression for it. These subarrangements,
called {\em $2$-structures}, were introduced
by Herb~\cite{Herb-2S} to study characters of discrete
series of real reductive groups.
In fact, the identity of~\cite[Theorem~3.2.1]{EMR} is valid for
any valuation and its proof uses
only basic properties of Coxeter systems and
closed convex polyhedral cones. 

In this paper we use the setting of $2$-structures 
and~\cite[Corollary~3.2.4]{EMR} (recalled in
\cref{thm_pizza_abstract})
to obtain a dissection
proof of the higher-dimensional
pizza theorem of~\cite[Corollary~6.9]{EMR_pizza}
that is independent of the results and methods of~\cite{EMR_pizza}
(and of~\cite{Brailov}):

\begin{theorem}[Abstract pizza theorem;
see \cref{theorem_pizza_dissection}.]
With the notation and hypotheses of \cref{theorem_introduction_first},
we have
\[\sum_{T}(-1)^T[\overline{T}\cap(K+a)] = 
\sum_{T}(-1)^T[T\cap(K+a)] =
0,\]
where the brackets denote classes in the abstract dissection
group of \cref{def_K(V)}.
\label{theorem_introduction_second}
\end{theorem}

As we take into account lower-dimensional sets when defining our abstract
dissection group, this result implies
generalizations of the higher-dimensional pizza theorem to all the
intrinsic volumes (if $K$ is convex).
The idea of the proof of \cref{theorem_introduction_second}
is the following:
by expanding the expression using $2$-structures, we can
reduce to a sum where each term is a similar expression for
an arrangement that is a product of arrangements
of types~$A_1$ and~$I_2(2^k)$. We then adapt the dissection proof
of Frederickson to an arrangement
of type $I_2(2m)\times\Hf'$. We also explain how to keep track of 
lower-dimensional regions of the dissection.
If our product arrangement contains at least
one dihedral factor, then its contribution is zero, and we immediately
get a dissection proof of the result.
However, if all the product arrangements
that appear are of type $A_1^n$, then their individual contributions are not zero.
We need one extra step in the proof
to show that the contributions cancel.
This step uses a slight
refinement of the Bolyai-Gerwien theorem presented
in \cref{section_Bolyai-Gerwien_plus_epsilon}.
An interesting point to note is that the shape of the pizza plays
absolutely no role in this proof, as long as it has 
the same symmetries as the arrangement and contains
the convex hull of $\{w(a) : w \in W\}$. In particular, we no longer
need to assume that it is measurable and of finite volume.



Let us mention some interesting questions that remain open:
\begin{itemize}
\item[(1)] The paper~\cite{EMR_pizza} proves the pizza
theorem for more general arrangements (the condition is that
the arrangement~$\Hf$ is a Coxeter arrangement
and that its number of hyperplanes is greater than the dimension $n$ and
has the same parity as that dimension), but only in the case of the ball;
see~\cite[Corollary~7.6]{EMR_pizza}.
Is it possible to give a dissection proof of this result?

\item[(2)] Mabry and Deiermann~\cite{MaDe} show that the
two-dimensional pizza theorem does not hold for a dihedral arrangement having
an odd number of lines. More precisely, they determine the sign
of the quantity $\sum_T(-1)^T\Vol(T\cap K)$, where $K$
is a disc containing the origin, and show that it vanishes if and only
if the center of $K$ lies on one of the lines. Their methods are analytic.
As far as we know, there exists no dissection proof of this result
either.
The higher-dimensional case where $\Hf$ is a Coxeter arrangement
and the number of its hyperplanes does not have the same parity as
$n$ also remains wide open.
\end{itemize}


\section{Review of 2-structures and of the basic identity}
\label{section_review}

Let $V$ be a finite-dimensional real vector space with an inner product
$(\cdot,\cdot)$. 
For every $\alpha\in V$, we denote by $H_\alpha$ the hyperplane
$\alpha^\perp$ and by $s_\alpha$ the orthogonal reflection in 
the hyperplane $H_\alpha$.

We say that a subset $\Phi$ of $V$ is a
\emph{normalized pseudo-root system} if:
\begin{itemize}
\item[(a)] $\Phi$ is a finite set of unit vectors;
\item[(b)] for all $\alpha,\beta\in\Phi$, we have $s_\beta(\alpha)\in\Phi$
(in particular, taking $\alpha=\beta$, we get that $-\alpha\in\Phi$).
\end{itemize}
Elements of $\Phi$ are called \emph{pseudo-roots}.
The \emph{rank} of $\Phi$ is the dimension of its span.

We call such objects {\em pseudo-root systems} to distinguish them from
the root systems that appear in representation theory.
If $\Phi'$ is a root system then  
$\Phi={\{\alpha/ {\|\alpha\|} : \alpha\in\Phi'\}}$ 
is a normalized pseudo-root system. 
Not every normalized pseudo-root system arises in this manner;
see for instance the pseudo-root systems of type $H_{3}$ and $H_{4}$.

We say that a normalized pseudo-root system $\Phi$ is \emph{irreducible}
if, whenever $\Phi=\Phi_1\sqcup\Phi_2$ with $\Phi_1$ and 
$\Phi_2$ orthogonal,
we have either $\Phi_1=\varnothing$ or $\Phi_2=\varnothing$. Every normalized
pseudo-root system can be written uniquely as a disjoint union of
irreducible normalized pseudo-root systems. Irreducible
normalized pseudo-root systems are classified: they are in one
of the infinite families $A_n$ ($n\geq 1$), $B_n/C_n$ ($n\geq 2$),
\footnote{The pseudo-root systems of types $B_n$ and $C_n$ are identical
after normalizing the lengths of the roots.}
$D_n$ ($n\geq 4$), $I_2(m)$ ($m\geq 3$) or one of the exceptional
types $E_6$, $E_7$, $E_8$, $F_4$, $H_3$ or $H_4$, with
types $I_2(3)$ and $A_2$ isomorphic, as well as types
$I_2(4)$ and $B_2$.
(See
\cite[Chapter~5]{GB}
or Table~1 in~\cite[Appendix~A]{BB}.)

We say that a subset $\Phi^+\subset\Phi$ is a
\emph{positive system} if there exists a total ordering
$<$ on the $\R$-vector space $V$ such that $\Phi^+=\{\alpha\in\Phi :
\alpha>0\}$ (see~\cite[Section~1.3]{Hu-Cox}).
The \emph{Coxeter group} of $\Phi$ is the group of isometries $W$ of
$V$ generated by the reflections $s_\alpha$ for $\alpha\in\Phi$.
This group preserves~$\Phi$ by definition of a normalized pseudo-root
system, and it acts simply transitively on the set of positive systems
by~\cite[Section~1.4]{Hu-Cox}. In particular, the Coxeter group $W$ is finite.

Let $E$ be a finite set of unit vectors of $V$ such that $E\cap(-E)=
\varnothing$. The corresponding \emph{hyperplane arrangement} is
the set of hyperplanes $\Hf=\{H_e : e\in E\}$. 
A \emph{chamber} of $\Hf$ is
a connected component
of $V - \bigcup_{e\in E}H_e$;
we denote by $\Tc(\Hf)$ the set of chambers of~$\Hf$.
Fix a chamber $T_0$ to be the {\em base chamber}.
For a chamber $T \in \Tc(\Hf)$
we denote by $S(T,T_0)$ the set of
$e \in E$ such that the two chambers $T$ and~$T_0$ are on different
sides of the hyperplane~$H_e$, and define the \emph{sign} of $T$
to be $(-1)^T=(-1)^{|S(T,T_0)|}$.

We say that
$\Hf$ is a \emph{Coxeter arrangement} if it is stable by 
the orthogonal reflections in
each of its hyperplanes. In that case, the set $\Phi=E\cup(-E)$ is a
normalized pseudo-root system. We call its Coxeter group the Coxeter group
of the arrangement. 
The map sending a positive
system $\Phi^+\subset\Phi$ to the set
$\{v\in V : \forall\alpha\in\Phi^+ \: (v,\alpha) > 0\}$
is a bijection from the positive systems in $\Phi$ to the chambers
of $\Hf$. See for example
\cite[Chapitre~V \S~4 No.~8 Proposition~9 p.~99]{Bourbaki}
and the discussion following it.
Conversely, if $\Phi\subset V$ is a normalized pseudo-root
system with Coxeter group $W$
and $\Phi^+\subset\Phi$ is a positive system, then
$\Hf=\{H_\alpha : \alpha\in\Phi^+\}$ is a Coxeter hyperplane
arrangement, and in that case we always take the base chamber~$T_0$ to be
the chamber corresponding to $\Phi^+$.

Let us define product arrangements. Let $V_1$ and $V_2$ be two
finite-dimensional real vector spaces equipped with inner products, and
suppose that we are given hyperplane arrangements $\Hf_1$ and
$\Hf_2$ on $V_1$ and $V_2$ respectively. We consider the product
space $V_1\times V_2$, where the factors are orthogonal.
The {\em product arrangement} $\Hf_1\times\Hf_2$ is then the arrangement on
$V_1\times V_2$ with hyperplanes $H\times V_2$ for $H\in\Hf_1$ and
$V_1\times H'$ for $H'\in\Hf_2$. 
If $\Hf_1$ is the empty arrangement, then we write
$V_1\times \Hf_2$  instead of the confusing $\varnothing \times\Hf_2$.
Similarly, if $\Hf_2$ is the empty arrangement, we write $\Hf_1\times V_2$.
If the arrangements
$\Hf_1$ and $\Hf_2$ arise from normalized pseudo-root systems
$\Phi_1\subset V_1$ and $\Phi_2\subset V_2$, then 
their product $\Hf_1\times\Hf_2$
arises from the normalized pseudo-root system
$\Phi_1\times\{0\}\cup\{0\}\times\Phi_2\subset V_1\times V_2$.
We also denote this pseudo-root system by $\Phi_1\times\Phi_2$.

The notion of $2$-structures was introduced by Herb
for root systems to study the characters of discrete series representations;
see for example the review article~\cite{Herb-2S}. The definition we
give here is Definition~B.2.1 of~\cite{EMR}.
It has been slightly adapted to work for pseudo-root systems.

\begin{definition}
Let $\Phi$ be a normalized pseudo-root system with Coxeter group $W$.
A \emph{$2$-structure} for $\Phi$ is a subset~$\varphi$ of~$\Phi$
satisfying the following properties:
\begin{itemize}
\item[(a)] 
The subset $\varphi$ is a disjoint union
$\varphi=\varphi_1\sqcup\varphi_2\sqcup\cdots\sqcup\varphi_r$,
where the $\varphi_i$ are pairwise orthogonal subsets
of $\varphi$ and each of them is an irreducible pseudo-root system of type
$A_1$, $B_2$ or $I_2(2^k)$ for $k\geq 3$.
\item[(b)] Let $\varphi^+=\varphi\cap\Phi^+$. If $w\in W$ is such that
$w(\varphi^+)=\varphi^+$ then $\det(w) = 1$.
\end{itemize}
\label{def_2_structure}
We denote by $\Tt(\Phi)$ the set of $2$-structures for $\Phi$.
\end{definition}

\begin{proposition}
Let $\Phi$ be a normalized pseudo-root system with Coxeter group
$W$.
\begin{itemize}
\item[(i)] The group $W$ acts transitively on
the set of $2$-structures $\Tt(\Phi)$.

\item[(ii)] The pseudo-root system $\Phi$ and its $2$-structures
have the same rank if and only if there exists $w\in W$ whose
restriction to $\Span(\Phi)$ is equal to
$-\id_{\Span(\Phi)}$.

\end{itemize}
\label{proposition_2_structures}
\end{proposition}


To each $2$-structure $\varphi\subset\Phi$, we can associate a
sign $\epsilon(\varphi)=\epsilon(\varphi,\Phi^+)$
(see the start of Section~5 and Lemma~5.1 of~\cite{Herb-DSC}
and Definition~B.2.8 of~\cite{EMR}).

We next introduce the abstract pizza quantity.
Let $\Hf$ be a central hyperplane arrangement on~$V$.
Let $\Cf_\Hf(V)$ be the set of closed convex polyhedral cones in~$V$
that are intersections of closed half-spaces bounded by hyperplanes~$H$
where $H\in\Hf$,
and let $K_\Hf(V)$ be the quotient of the free abelian group 
$\bigoplus_{K\in \Cf_\Hf(V)}\Z[K]$
on $\Cf_\Hf(V)$ by the relation
$[K]+[K']=[K\cup K']+[K\cap K']$ for all
$K,K'\in \Cf_\Hf(V)$ such that $K\cup K'\in \Cf_\Hf(V)$.
For $K\in \Cf_\Hf(V)$, we still denote
the image of~$K$ in $K_\Hf(V)$ by~$[K]$.
\begin{definition}
Suppose that we have fixed a base chamber of $\Hf$.
The \emph{abstract pizza quantity} of~$\Hf$ is
\[P(\Hf)=\sum_{T\in\Tc(\Hf)}(-1)^T[\overline{T}]\in K_\Hf(V).\]
\end{definition}
\begin{remark}
By Lemma~3.2.3 of~\cite{EMR}, we have
\[P(\Hf)=\sum_{T\in\Tc(\Hf)}(-1)^T[T].\]
We use this alternative definition of $P(\Hf)$ in our proofs.
\label{remark_open_chambers}
\end{remark}




The following result is Corollary~3.2.4 of~\cite{EMR}.
It shows how to evaluate the pizza quantity for a Coxeter arrangement
in terms of the associated $2$-structures.
\begin{theorem}
Let $\Phi\subset V$ be a normalized pseudo-root
system. Choose a positive system $\Phi^+\subset\Phi$
and let $\Hf$ be the hyperplane
arrangement $(H_\alpha)_{\alpha\in\Phi^+}$ on $V$ with base chamber
corresponding to the positive system~$\Phi^+$.
For every $2$-structure $\varphi\in\Tt(\Phi)$, we 
write $\varphi^+=\varphi\cap\Phi^+$ and
we denote
by $\Hf_\varphi$ the hyperplane arrangement $(H_\alpha)_{\alpha\in\varphi^+}$
with base chamber corresponding to $\varphi^+$.
Then we have
\[P(\Hf)=\sum_{\varphi\in\Tt(\Phi)}\epsilon(\varphi)
P(\Hf_\varphi).\]

\label{thm_pizza_abstract}
\end{theorem}

If $\varphi\in\Tt(\Phi)$ then the closures of the chambers of $\Hf_\varphi$ 
are elements of $\Cf_\Hf(V)$, so $P(\Hf_\varphi)$ makes
sense as an element of $K_\Hf(V)$.

\section{A dissection proof of the 
higher-dimensional pizza theorem}
\label{section_dissection}

\begin{definition}
Let $\Cf(V)$ be a non-empty family of
subsets of $V$ that is stable by finite intersections and affine
isometries and such that, if $C\in \Cf(V)$ and
$D$ is a closed affine half-space of $V$, then $C\cap D\in \Cf(V)$.
Furthermore, we assume that $\Cf(V)$ is
closed with respect to Cartesian products, that is,
if $C_{i} \in \Cf(V_{i})$ for $i=0,1$ then 
$C_{0} \times C_{1} \in \Cf(V_{0} \times V_{1})$.
For example, we could take $\Cf(V)$ to be the family of all
convex subsets of $V$, or of all closed (or compact) convex subsets, or of all
convex polyhedra.

We denote by $K(V)$ the quotient of the free abelian group
$\bigoplus_{C\in \Cf(V)}\Z[C]$ on $\Cf(V)$ by the three relations:
(i) $[\varnothing]=0$;
(ii) $[C\cup C']+[C\cap C']=[C]+[C']$ for all $C,C'\in \Cf(V)$ such that
$C\cup C'\in \Cf(V)$;
and
(iii) $[g(C)]=[C]$, for every $C\in \Cf(V)$ and every affine
isometry $g$ of $V$.
For $C\in \Cf(V)$, we still denote the image of $C$ in $K(V)$ by
$[C]$. 
\label{def_K(V)}
\end{definition}

A \emph{valuation} on $\Cf(V)$ with values in an abelian group $A$ is
a function $\Cf(V)\fl A$ that can be extended to a morphism
of groups $K(V)\fl A$.

\begin{remark}
Define $\Bf(V)$ to be the relative Boolean algebra
generated by $\Cf(V)$, that is, the smallest collection
of subsets of $V$ that contains $\Cf(V)$ and is closed
under finite unions, finite intersections and set differences.
Groemer's Integral Theorem states that
a valuation on $\Cf(V)$ can be extended
to a valuation on the Boolean algebra $B(V)$; see~\cite{Gr}
and also~\cite[Chapter~2]{Klain_Rota}.
Applying this to the valuation $C \longmapsto [C]$ with values in $K(V)$,
we see that we can make sense of~$[C]$ for any $C\in \Bf(V)$.
For instance, we have
$[C_1 \cup C_2] = [C_1] + [C_2] - [C_1 \cap C_2]$
and
$[C_1 - C_2] = [C_1] - [C_1 \cap C_2]$.
Moreover if $\Cf(V)$ is the set of all convex polyhedra in
$V$, then $\Bf(V)$ contains all polyhedra (convex or not), and also
half-open polyhedra.
\label{remark_[C]_for_more_general_C}
\end{remark}

Next we have the following straightforward lemma, whose proof we omit, 
which states that the class symbol is well-behaved with respect
to Cartesian products.
\begin{lemma}
The two class identities 
$[C_{0}] = [D_{0}]$ and $[C_{1}] = [D_{1}]$
in 
$K(V_{0})$ and~$K(V_{1})$ respectively
imply that
$[C_{0} \times C_{1}] = [D_{0} \times D_{1}]$
in
$K(V_{0}) \times K(V_{1})$.
\label{lemma_product}
\end{lemma}



Let $\Hf$ be a central hyperplane arrangement on $V$ with fixed
base chamber.
If $K\in \Cf(V)$,
we have a morphism of groups $e_K:K_\Hf(V)\longrightarrow K(V)$ 
induced by the map $\Cf_\Hf(V) \longrightarrow \Cf(V)$, $C\longmapsto C\cap K$.

We denote by $P(\Hf,K)$ 
the image of $P(\Hf)$ by this morphism $e_K$;
in other words, we have
\begin{align*}
P(\Hf,K)
& =
\sum_{T\in\Tc(\Hf)}(-1)^T[\overline{T}\cap K]
=
\sum_{T\in\Tc(\Hf)}(-1)^T[T\cap K],
\end{align*}
where \cref{remark_open_chambers} implies the second equality.



We state the main theorem of this paper.
First for $u,v \in V$ define the half-open line segment
$(u,v]$ by $\{(1-\lambda) u + \lambda v : 0 < \lambda \leq 1\}$.
Our main result is the following:
\begin{theorem}
[The Abstract Pizza Theorem]
Let $\Hf$ be a Coxeter hyperplane arrangement with Coxeter group~$W$
in an $n$-dimensional space $V$ such that $-\id_V\in W$.
Let $K\in \Cf(V)$ and $a\in V$. Suppose that $K$ is stable by the group~$W$
and contains the convex hull of the set $\{w(a) : w\in W\}$.
\begin{enumerate}
\item If $\Hf$ is not of type $A_1^{n}$, 
we have $P(\Hf,K+a)=0$ in $K(V)$.

\item If $\Hf$ has type $A_1^{n}$, $\Phi$ is the normalized
pseudo-root system corresponding to~$\Hf$ and 
$\Phi^+=\{e_1,\ldots,e_n\}$ where $\Phi^+\subset\Phi$ is the positive
system corresponding to the base chamber of~$\Hf$,
then the following identity holds:
\begin{align}
\label{equation_half_open}
P(\Hf,K+a)
& =
\left[\prod_{i=1}^n(0,2(a,e_i)e_i]\right] .
\end{align}
\end{enumerate}
\label{theorem_pizza_dissection}
\end{theorem}
Here we are using \cref{remark_[C]_for_more_general_C} to make
sense of the right-hand side of equation~\eqref{equation_half_open}.


The conditions on $K$ are satisfied if for example $K$ is convex,
contained in $\Cf(V)$, stable by $W$ and $0\in K+a$. Indeed, the last condition
implies that $-a\in K$; as
$-\id_V\in W$ by assumption, this in turns implies that $a\in K$,
hence that $K$ contains the convex hull of the set $\{w(a) : w \in W\}$.

We will give the proof of \cref{theorem_pizza_dissection} at the end of the section. This proof does not use Corollary~6.9 of~\cite{EMR_pizza}, so we obtain
a new proof of that result.


Let $V_0,\ldots,V_n$ denote the intrinsic volumes on $V$
(see \cite[Section~4.2]{Schneider}).

\begin{lemma}
Let $(v_1,\ldots,v_k)$ be an orthogonal family of vectors in $V$. Then
$V_i((0,v_1]\times\cdots\times(0,v_k]) = 0$
for $0 \leq i \leq k-1$.
\label{lemma_intrinsic_volume_half_open_parallelotope}
\end{lemma}


\begin{corollary}
We keep the notation and hypotheses of \cref{theorem_pizza_dissection}.
If $\Hf$ is not of type $A_1^n$, we have
\begin{equation}
\sum_{T\in\Tc(\Hf)}(-1)^T V_i(T\cap(K+a))=0
\label{equation_intrinsic_volumes}
\end{equation}
for every $0 \leq i \leq n$, where $K$ is assumed to be convex if
$i\not=n$.
If $\Hf$ has type $A_1^n$ and $K$ is convex
then equation~\eqref{equation_intrinsic_volumes}
holds for $0 \leq i \leq n-1$.
\end{corollary}





\begin{remark}
\cref{theorem_pizza_dissection} immediately implies
generalizations to our higher-dimensional case
of the ``thin crust'' and ``thick crust'' results
of Confection~3 and Leftovers~1 of~\cite{MaDe}
for an even number of cuts.

We obtain the ``thin crust'' result by evaluating the
$(n-1)$st intrinsic volume on $P(\Hf,K+a)$. Note that
this result holds for a pizza of any (convex) shape and even
in the case where we only make $n$ cuts, where $n$ is the
dimension.

To generalize the ``thick crust'' result, consider two sets
$K\subset L$ stable by $W$ and
in~$\Cf(V)$. If $a\in V$
is such that $K$ contains the convex hull of the set $\{w(a) : w\in W\}$,
then
\[P(\Hf,(L - K)+a)=P(\Hf,L+a)-P(\Hf,K+a)=0,\]
so in particular
\[\sum_{T\in\Tc(\Hf)}(-1)^T\Vol(T\cap((L - K)+a))=0.\]
The case where $K$ and $L$ are balls with the same center
is the ``thick crust'' result.
\label{remark_crust}
\end{remark}


We now state 
some lemmas that will be used in 
the proof of \cref{theorem_pizza_dissection}.

\begin{lemma}
Let $\Hf_i$ be a hyperplane arrangement on $V_i$ for $i=0,1$.
Assume furthermore that $\Hf_1 = \{H_e\}_{e \in E_1}$ has type $A_1^r$
and $\dim(V_1) = r$.
Let $E_{1} = \{e_1, \ldots, e_r\}$ be the index set of $\Hf_{1}$.
Let $\Hf$ and $V$ be the Cartesian products $\Hf_0 \times \Hf_{1}$ and
$V_0 \times V_1$ respectively.
Then for every $K\in \Cf(V)$
that is stable under the orthogonal reflections in the hyperplanes
$V_0 \times H_{e_1}, \ldots, V_0 \times H_{e_r}$ and for every $a\in V_1$, if
$L=K+a$, we have the identity
\begin{align*}
P(\Hf,L)
& =
P\left(\Hf_0\times V_1,L\cap (V_0\times(0,2(a,e_1)e_1]\times\cdots\times(0,2(a,e_r)e_r])\right),
\end{align*}
where $\Hf_0\times V_1$ is the product of
$\Hf_0$ and the empty hyperplane arrangement on $V_1$.
\label{lemma_Cartesian_product_with_A_1}
\end{lemma}


\begin{lemma}
Suppose that we have $V=V_1^{(1)}\times\cdots\times V_1^{(r)}\times V_2^{(1)}\times
\cdots\times V_2^{(s)}$, where the factors of the product are pairwise
orthogonal, and that $\Hf$ is a product
$\Hf_1^{(1)}\times\cdots\times\Hf_1^{(r)}\times\Hf_2^{(1)}\times\cdots
\times\Hf_2^{(s)}$, where each $\Hf_i^{(j)}$ is a hyperplane arrangement
on $V_i^{(j)}$. Suppose further that:
\begin{itemize}
\item[(a)] If $1 \leq j \leq r$ then $V_1^{(j)}$ is $1$-dimensional, and
we have a unit vector $e^{(j)}$ in $V_1^{(j)}$
yielding the hyperplane arrangement $\Hf_1^{(j)} = \{0\}$.

\item[(b)] If $1 \leq j \leq s$ then $V_2^{(j)}$ is $2$-dimensional, and
the arrangement $\Hf_2^{(j)}$ is of type $I_2(2m^{(j)})$ for some
$m^{(j)} \geq 2$.
\end{itemize}
Let $a\in V$ and $K\in \Cf(V)$.
Suppose that $K$ is stable by the Coxeter group $W$ and contains the convex
hull of the set $\{w(a) : w\in W\}$.
Then the following two statements hold:
\begin{itemize}
\item[(i)] If $s\geq 1$ we have $P(\Hf,K+a)=0$ in $K(V)$.

\item[(ii)] If $s=0$ we have in $K(V)$ the identity
\begin{align*}
P(\Hf,K+a)
& =
\left[(0,2(a,e^{(1)})e^{(1)}]\times\cdots\times(0,2(a,e^{(r)})e^{(r)}]\right] .
\end{align*}
\end{itemize}
\label{lemma_pizza_for_A_1^rxI_2^s}
\end{lemma}




\begin{proof}[Proof of \cref{theorem_pizza_dissection}.]
Statement~(ii) is exactly \cref{lemma_pizza_for_A_1^rxI_2^s}(ii).
We now prove statement~(i), so we assume that $\Hf$ is not of type
$A_1^{n}$.
By \cref{thm_pizza_abstract}, we have
\[P(\Hf,K+a)=\sum_{\varphi\in\Tt(\Phi)}\epsilon(\varphi)
P(\Hf_\varphi,K+a) . \]
By definition, any $2$-structure for $\Phi$ is of
type $A_1^r\times\prod_{k\geq 2}I_2(2^k)^{s_k}$
with
$\sum_{k\geq 2}s_k$ finite
and, as $W$ acts transitively on the set
of $2$-structures 
(\cref{proposition_2_structures}(i)),
the integers $r$ and $s_k$, for $k\geq 2$, do not
depend on the $2$-structure but only on $\Phi$.
Also, by \cref{proposition_2_structures}(ii), 
we have $\dim V=r+\sum_{k\geq 2}2s_k$, so we are in the situation
of \cref{lemma_pizza_for_A_1^rxI_2^s}.
Suppose that $\sum_{k\geq 2}s_k\geq 1$. 
Then by \cref{lemma_pizza_for_A_1^rxI_2^s}(i)
we have $P(\Hf_\varphi,K+a)=0$ for every $\varphi\in\Tt(\Phi)$
and hence $P(\Hf,K+a)=0$.
Assume now that $\sum_{k\geq 2}s_k=0$, that is, $s_{k} = 0$ for every $k$.
Statement~(ii) of the same lemma implies that
\begin{align}
P(\Hf,K+a)=\sum_{\varphi\in\Tt(\Phi)}\epsilon(\varphi)
\Bigg[\prod_{e\in\varphi\cap\Phi^+}(0,2(a,e)e]\Bigg].
\label{equation_sunshine}
\end{align}
This is an alternating sum of classes of half-open
rectangular parallelotopes in $V$.
So we can apply \cref{thm_Bolyai-Gerwien_plus_epsilon} to
prove that $P(\Hf,K+a)=0$ in $K(V)$. We know that
$V_i(P(\Hf,K+a))=0$ if $0\leq i\leq n-1$ by
\cref{lemma_intrinsic_volume_half_open_parallelotope},
so it remains to prove that $V_n(P(\Hf,K+a))=0$, that is,
that the alternating sum of the volumes of the parallelotopes
$\prod_{e\in\varphi\cap\Phi^+}(0,2(a,e)e]$ is equal to zero.
This follows from
Corollary~6.9 of the paper~\cite{EMR_pizza}.
However we can now give a direct
proof (that does not use analysis)
using the method of that corollary. 
Let $f:V\fl\R$ be the function defined by
\begin{align*}
f(a)
& =
\sum_{\varphi\in\Tt(\Phi)}\epsilon(\varphi) \prod_{e\in\varphi\cap\Phi^+} 2(a,e) .
\label{equation_sunshine}
\end{align*}
Note that $f$ is a polynomial homogeneous of degree $n$ on~$V$.
Furthermore equation~\eqref{equation_sunshine}
implies that
$\Vol(P(\Hf,K+a))  
=f(a)$
for every convex subset $K$ of $V$ of finite volume
that is stable by $W$ and every $a\in V$ such that $0\in K+a$.
The polynomial~$f$ satisfies $f(w(a))=\det(w)f(a)$ for every
$w\in W$ and every $a\in V$ (this is easy to see;
see for example Corollary~2.3 of~\cite{EMR_pizza}),
so it vanishes on every hyperplane of~$\Hf$. But if $f\not=0$, then
the vanishing set of $f$ must be of degree at most~$n$, 
which contradicts
the fact that, as $\Hf$ is not of type~$A_1^{n}$, we have $|\Hf|>n$.
Hence we must have $f=0$, and this gives the desired result.
\end{proof}



\section{The Bolyai-Gerwien theorem for parallelotopes}
\label{section_Bolyai-Gerwien_plus_epsilon}

The classical Bolyai-Gerwien theorem states that two polygons are
scissors congruent if and only if they have the same area. There is also
a well-known generalization in higher dimensions that applies
to parallelotopes; it follows from the characterization of translational
scissors congruences in arbitrary dimensions, and was proved independently
by Jessen-Sah and Sah, see 
the beginning of Section~7 of~\cite{Jessen_Thorup} or
Theorem~1.1 in Chapter~4 of~\cite{Sah}. 
In this section, we state a slight refinement of this
generalization, \cref{thm_Bolyai-Gerwien_plus_epsilon},
that keeps track of lower-dimensional
faces; in other words, we do not want ignore the boundaries.


As in the previous sections, let $V$ be an $n$-dimensional real
vector space with an inner product $(\cdot,\cdot)$.
If $(v_1,\ldots,v_r)$ is a linearly independent list of elements of $V$,
we define the parallelotope
$P(v_1,\ldots,v_r)=\left\{\sum_{i=1}^r a_i v_i : 0\leq a_i\leq 1\right\}$.
We denote by
$\Pf(V)$ the set of all convex polytopes in $V$ (including lower-dimensional
ones) and by
$\Zf(V)$ the subfamily of polytopes that are translates of parallelotopes
of the form $P(v_1,\ldots,v_r)$.
The set $\Pf(V)$ satisfies the
conditions of \cref{def_K(V)}, 
so we can define an abelian group
$K_\Pf(V)$ as in that definition. We denote by
$K_\Zf(V)$ the subgroup of $K_\Pf(V)$ generated by the classes~$[P]$
for $P\in\Zf(V)$.
\cref{remark_[C]_for_more_general_C} implies that, if
$\Pfe(V)$ is 
the relative Boolean algebra generated
by $\Pf(V)$, 
then we can define the class
$[P]$ in $K_\Pf(V)$
of any element $P$ in
$\Pfe(V)$.
We denote by $\Zfe(V)$ the set of elements $P$ of $\Pfe(V)$ such that
$[P]\in K_\Pf(V)$ is in the subgroup $K_\Zf(V)$. For example,
the set $\Zfe(V)$ contains $\Zf(V)$, and it also contains all half-open
parallelotopes.

Let $V_0,\ldots,V_n$ be the intrinsic volumes on $V$;
see~\cite[Section~4.2]{Schneider}. These are valuations on the set of
all compact convex subsets of $V$, and in particular on
$\Pf(V)$, so they induce morphisms of
groups from $K_\Pf(V)$ to $\R$, that we still denote
by $V_0,\ldots,V_n$. Note that $V_0$ is the Euler-Poincar\'e characteristic
with compact support,
so the image of $K_\Pf(V)$ is~$\Z$.

The main result of this section is the following isomorphism.

\begin{theorem}
The morphism $(V_0,V_1,\ldots,V_n):K_\Zf(V)\fl\Z\times\R^n$ is an
isomorphism.
In particular, if $P,P'\in
\Zfe(V)$ are such that $V_i(P)=V_i(P')$ for every $0 \leq i \leq n$,
then $[P]=[P']$ in~$K_\Zf(V)$.
\label{thm_Bolyai-Gerwien_plus_epsilon}
\end{theorem}




\acknowledgements{The authors thank Dominik Schmid
for introducing them to the Pizza Theorem, and Ramon van Handel
for dispelling some of their misconceptions and lending them
a copy of \cite{Schneider}.
They made extensive use of Geogebra to understand the $2$-dimensional
situation.}



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



\end{document}

