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%% define your title in the usual way
\title{Dual mixed volumes of polytopes}

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\author[Y. Gao, T. Lam, and L. Xue]{\texorpdfstring{Yibo Gao\thanks{Yibo Gao was partially supported by NSFC Grants No. 12471309. \href{mailto:gaoyibo@bicmr.pku.edu.cn}{gaoyibo@bicmr.pku.edu.cn}}\addressmark{1}, Thomas Lam\thanks{Thomas Lam was partially supported by Grant No. DMS-1953852 and DMS-2348799 from the National Science
Foundation. \href{mailto:tfylam@umich.edu}{tfylam@umich.edu}}\addressmark{2}, \and Lei Xue\thanks{\href{mailto:leixue@umich.edu}{leixue@umich.edu}}\addressmark{3}}{Yibo Gao, Thomas Lam, and Lei Xue}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1} Beijing International Center for Mathematical Research, Peking University, \mbox{Beijing,} China \\ \addressmark{2,3} Department of Mathematics, University of Michigan, \mbox{Ann Arbor, MI} USA}

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\abstract{We define and study the dual mixed volume rational function of a sequence of polytopes, a dual version of the mixed volume polynomial. This concept has direct relations to the adjoint polynomials and the canonical forms of polytopes. We show that dual mixed volume is additive under mixed subdivisions, and is related by a change of variables to the dual volume of the Cayley polytope.  We study dual mixed volume of zonotopes, generalized permutohedra, and associahedra.  The latter reproduces the planar $\phi^3$-scalar amplitude at tree level.}

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\keywords{Polytopes, mixed volume, canonical form, mixed subdivisions, scattering amplitude}

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\begin{document}
\maketitle
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\title{Dual mixed volume}

\section{Introduction}\label{sec:intro}






Let $\Vol(\cdot)$ be the normalized volume function in $\R^d$ where the coordinate simplex has volume $1$. For a sequence of convex bodies $\S =(S_1,S_2,\ldots,S_r)$ in $\R^d$ and $x_1,\ldots,x_r >0$, 
\begin{equation}\label{eq:MV}
\Vol_\S(\x) := \Vol(x_1 S_1 + \cdots + x_r S_r) = \sum_{i_1,i_2,\ldots,i_d=1}^r V(S_{i_1},S_{i_2},\ldots,S_{i_d}) x_{i_1} \cdots x_{i_d},
\end{equation}
where $V(S_{i_1},S_{i_2},\ldots,S_{i_d})$ are the \emph{mixed volumes} of $S_{i_1},\ldots,S_{i_d}$, and we call $\Vol_\S(\x)$ the \emph{mixed volume polynomial}.  
\begin{comment}
The mixed volumes are characterized by the following fundamental properties.
\begin{enumerate}
\item $V(S,S,\ldots,S) = \Vol(S)$,
\item $V$ is symmetric in its arguments,
\item $V$ is multilinear: $V(aS+bS',S_2,\ldots,S_d) = aV(S,S_2,\ldots,S_d)+b V(S',S_2,\ldots,S_d)$.
\end{enumerate}
\end{comment}
Mixed volumes are nonnegative and satisfy various inequalities, among them the \emph{Alexandrov--Fenchel inequality}:
\begin{equation}\label{eq:AF}
V(S_1,S_2,S_3,\ldots,S_d)^2 \geq V(S_1,S_1,S_3,\ldots,S_d) \cdot V(S_2,S_2,S_3,\ldots,S_d),
\end{equation}
a deep result in the geometry of convex bodies, with exciting applications in combinatorics (see \cite{Schneider,stanley-combinatorial-Aleksandrov-Fenchel}).

In this work, we study the \emph{dual mixed volume function}, defined by
$$
m_\S (\x) = m_\S (x_1,\ldots,x_r): = \Vol((x_1S_1 + \cdots + x_r S_r)^\vee) 
$$
where $S^\vee$ denotes the polar of a convex set $S$.  For general convex bodies, $m_\S(\x)$ is a complicated analytic object, but for a sequence of polytopes $\P = (P_1,\ldots,P_r)$, the dual volume function $m_\P(\x)$ is a rational function.  We focus on polytopes in this work, leaving the general dual mixed volume function for future investigation.

Besides the natural parallel with the mixed volume polynomial, the dual mixed volume function is motivated by recent developments in \emph{positive geometry} \cite{ABL,lam2022invitation}, occurring at the interface of combinatorial algebraic geometry and the physics of scattering amplitudes.  The dual mixed volume function specializes to the \emph{canonical form} of a polytope and produces in a special case the scalar $\phi^3$-amplitude (see \Cref{sec:associahedra}), the field theory limit of the open string amplitude.

This extended abstract is organized as follows. In \Cref{sec:dual-volume}, we define the \emph{dual volume} $\VolC(P)$ and the \emph{dual volume function} $\VolC_{\z}(P)$ of a polyhedron $P$ in a rigorous way, including those polyhedra that do not contain the origin in its interior, whose polar duals are unbounded. In \Cref{sec:properties}, we show that $\VolC(P)$ and $\VolC_{\z}(P)$ are valuative, generalizing results of Filliman \cite{filliman} and Kuperberg \cite{Kuperburg}. We also illustrate that the numerator of $\VolC_{\z}(P)$, suitably normalized, coincides with the \emph{adjoint polynomial} of (the cone of) the dual polytope, originally defined by \cite{Warren}, and that the \emph{canonical form}, in the sense of positive geometry, is given by $\Omega(P)=\VolC_{\z}(P)dz_1\cdots dz_d$. In \Cref{sec:Minkowski}, we define our main object of study, the \emph{dual mixed volume function} $m_{\P}(\x,\z)$ for a sequence of polytopes $\P=(P_1,\ldots,P_r)$ in $\R^d$. We provide formulae for the dual mixed volumes via integrals, mixed subdivisions and Cayley polytopes (\Cref{sec:Cayley}). The most important use cases of our study are \emph{generalized permutohedra} \cite{postnikov-permutohedra}. We first setup necessary tools to discuss polytopes that live in an affine hyperplane (\Cref{sec:affine}). Our results lead to curious identities on generalized permutohedra that are omitted here\footnote{A full version of the work summarized in this extended abstract appears in \cite{arxiv-version}.}. Specific cases of generalized permutohedra include zonotopes (\Cref{sec:zonotope}) and associahedra (\Cref{sec:associahedra}). In particular, we relate the dual mixed volume of an associahedron to the planar $\phi^3$-amplitude.

%In \Cref{sec:canform}, we show that the canonical form $\Omega(P)$ of a polytope $P$, in the sense of positive geometry, is given by $\Omega(P) = \Vol_\z(P) dz_1 \cdots dz_d$. As a consequence, we recover the fact that the adjoint hypersurface is the zero set of the canonical form (\Cref{cor:adjointzero}).  We show that the dual volume function behaves well under projective transformations (\Cref{cor:proj}).

%\subsection{Mixed subdivisions}


%We consider dual mixed volume functions for some important examples.  In \Cref{sec:affine}, we develop a formalism to discuss dual volume functions when the affine span of the polytope $P = P_1+ P_2+ \dots +P_r$ belongs to a hyperplane, which is often the case in examples.



%In \Cref{sec:genperm}, we consider \emph{generalized permutohedra}, polytopes obtained by parallel translating the facet hyperplanes of a permutohedron.  We give an explicit formula (\Cref{prop:genperm}) for the dual mixed volume of a generalized permutohedron.  The combinatorics of fine mixed subdivisions of generalized permutohedra has been studied in \cite{postnikov-permutohedra}.  Our results lead to some curious identities (\Cref{cor:Jm1} and \Cref{cor:Jm2}) that appear to be new.

%In \Cref{sec:associahedra}, we give an explicit formula (\Cref{prop:assoc}) for the dual mixed volume of an \emph{associahedron} and relate it to the planar $\phi^3$-amplitude.


\section{Definitions}\label{sec:dual-volume}


The main goal of this section is to define the \emph{dual volume} $\VolC(P)$ of a polyhedron $P \subseteq \R^d$, but we will start with some basic preliminaries. 


%\subsection{Cones, fans, and polyhedra}
A \emph{polyhedral cone} $C \subseteq \R^d$ is a non-empty intersection of finitely many closed half spaces, each passing through the origin $\mathbf{0}$.  A polyhedral cone $C$ is \emph{pointed} if it does not contain any line.  A \emph{face} of $C$ is an intersection of $C$ with a supporting hyperplane. A \emph{(polyhedral) fan} $\mathcal{F}$ is a finite set of polyhedral cones such that if $C\in\mathcal{F}$ and $F\subseteq C$ is a face, then $F \in\mathcal{F}$, and that if $C_1,C_2\in\mathcal{F}$, then $C_1\cap C_2$ is a face of both $C_1$ and $C_2$. A fan $\F$ in $\R^d$ is called \emph{complete} if the union of cones in it is the whole space $\R^d$.  
%We say that a polyhedral fan $\F$ is \emph{pure} of dimension $r$ if all maximal (under inclusion) cones of $\F$ are of dimension $r$.

A \emph{polyhedron} $P \subseteq \R^d$ is a non-empty intersection of finitely many closed half spaces (not necessarily passing through the origin).  A polyhedron $P$ is a \emph{polytope} if it is a bounded subset of $\R^d$.

For any non-empty closed convex set $S \subseteq\R^d$, the \emph{support function} of $S$ describes the (signed) distances from its supporting hyperplanes to the origin. It is given by
\begin{eqnarray}\label{eq:support}
h_S:\R^d &\rightarrow &\R \cup \{\infty\} \nonumber \\ 
\v &\mapsto & -\min_{\y\in S}\langle\v,\y\rangle.
\end{eqnarray}
For a polyhedron $P$, the \emph{normal fan} $\N(P)$ consists of the cones 
$$
C_F:=\{\v\in\R^d\:|\: h_P(\v)=-\langle\v,\y\rangle \neq \infty \text{ for every }\y\in F\}
$$ for each face $F$ of $P$. In particular, $h_P$ is a linear function on each $C_F$.  
%The support of $\N(P)$ is equal to the support of $h_P$.  
When $P$ is a polytope, its normal fan $\N(P)$ is a complete fan. %, i.e. $\R^d$ equals the union of all cones in $\N(P)$.

\begin{defin}\label{def:universal-dual-volume}
%Let $\mathcal{F}$ be a polyhedral fan in $\R^d$ pure of dimension $d$. Each $1$-dimensional face is called a \emph{extreme ray} in $\mathcal{F}$. For each extreme ray, we pick a vector $\mathbf{v}_i$ that spans it ($\mathbf{v}_i$ is called a \emph{generator} of the ray), and collect these vectors as $\mathbf{v}=(\mathbf{v}_1,\ldots,\mathbf{v}_g)$ and associate variables $u_1,\ldots,u_g$ correspondingly. 
Let $\mathcal{F}$ be a polyhedral fan generated by $g$ rays $\mathbf{v}_1,\ldots,\mathbf{v}_g$, and let $\mathcal{T}$ be any triangulation of $\mathcal{F}$ into full-dimensional \emph{simplicial cones} without adding generating rays, i.e., cones whose generators are linearly independent. Define the following rational function \[f_{\mathcal{F},\mathbf{v},\mathcal{T}}(u_1,\ldots,u_g):=\sum_{C=\mathrm{span}_{\R_{\geq0}}(\mathbf{v}_{j_1},\ldots,\mathbf{v}_{j_d})\in \mathcal{T}}\frac{|\det(\mathbf{v}_{j_1},\ldots,\mathbf{v}_{j_d})|}{u_{j_1}\cdots u_{j_d}}.\]
If $\F$ is a polyhedral fan in $\R^d$ pure of dimension $r < d$, then we define $f_{\F,\v,\T} := 0.$
\end{defin}

When $u_1,u_2,\ldots,u_g$ are all positive, $|\det(\mathbf{v}_{j_1},\ldots,\mathbf{v}_{j_d})|/(u_{j_1}\cdots u_{j_d})$ equals the normalized volume of the simplex formed by $\mathbf{0}$, $\mathbf{v}_{j_1}/u_{j_1}$, $\ldots$, $\mathbf{v}_{j_d}/u_{j_d}$.  While the formula for $f_{\mathcal{F},\mathbf{v},\mathcal{T}}$ may initially look daunting, it is simply the volume of some regions when the $u_i$-s are positive positive. We show that $f_{\mathcal{F},\mathbf{v},\mathcal{T}}$ does not depend on the triangulation $\mathcal{T}$. Therefore we write $f_{\mathcal{F},\mathbf{v}}(u_1,\ldots,u_g):=f_{\mathcal{F},\mathbf{v},\mathcal{T}}(u_1,\ldots,u_g)$ for any triangulation $\mathcal{T}$ of $\mathcal{F}$. 
\iffalse 
We also make the following definition for simplicity of notations.
\begin{defin}\label{def:universal-fan-dual-volume}
Let $h:\R^d\rightarrow\R$ be a function such that $h(c\x)=c\cdot h(\x)$ for all $\x\in\R^d$ and $c\in\R$. Let $\F$ be a polyhedral fan. Write \[f_{\F}(h):=f_{\F,\v}(u_1,\ldots,u_g)\] where $\v=(\v_1,\ldots,\v_g)$ are rays in $\F$ and $u_i=h(\v_i)$.
\end{defin}
Since $h(c\x)=c\cdot h(\x)$, scalings of $\v_i$'s do not matter and thus $f_{\F}(h)$ is well-defined. 
\fi

We say that $P$ is \emph{non-codegenerate} if the origin $\0$ is not contained in the affine span of any of the facets of $P$ and show that.  

\begin{lemma}
A polyhedron $P$ is non-codegenerate if and only if for any ray $\R_{\geq 0} \cdot \v$ of $\N(P)$, we have $h_P(\v) \neq 0$.
\end{lemma}


The following is the key definition of this section.
\begin{defin}\label{def:dual-volume}
Let $P\subseteq\R^d$ be a full-dimensional non-codegenerate polyhedron.
% such that $\mathbf{0}$ is not contained in the affine span of any facets of $P$, and let $\N(P)$ be its normal fan. 
For each ray in $\N(P)$, pick a vector that spans it, and collect them as $\mathbf{v}=(\mathbf{v}_1,\ldots,\mathbf{v}_g)$. Define the \emph{dual volume} \[\VolC(P):=f_{\N(P)}(h_{P})=f_{\N(P),\mathbf{v}}(h_P(\mathbf{v_1}),\ldots,h_P(\mathbf{v_g})).\] When $P$ is not full dimensional, set $\VolC(P)=0$. Also define the \emph{dual volume function}
 \[\VolC_{\z}(P):=\VolC(P-\z)\]
viewed as a rational function in the coordinates $z_1,\ldots,z_d$ of $\mathbf{z}$.
\end{defin}

%The following result is clear from the definitions.
\begin{lemma}
The dual volume $\VolC(P)$ and dual volume function $\VolC_{\z}(P)$ do not depend on the choice of $\mathbf{v}=(\mathbf{v}_1,\ldots,\mathbf{v}_g)$.
\end{lemma}

The name ``dual volume'' comes from its connection to the polar dual. Recall 
\begin{defin}\label{def:dual}
For a polyhedron $P\subseteq\R^d$, its \emph{polar dual} is \[P^\pol:=\{\v\in\R^d\:|\: h_P(\v)\leq 1\} = \{\v\in\R^d\:|\: \langle \v,\y \rangle \geq -1 \text{ for all } \y \in P\}. \]
\end{defin}
%\Yibo{We may need a proposition for polyhedra rather than polytopes as in draft2.}
\begin{prop}[{\cite[Chapter 4 (1.2)]{Barvinokconvexity}}]\label{prop:polar-dual-basics} 
If $P$ is a polytope with $\mathbf{0}$ in its interior, then $P^{\pol}$ is also a polytope with $\mathbf{0}$ in its interior. In this case $(P^{\pol})^{\pol}=P$. %and there is an order-reversing bijection on the face poset of $P$ and $P^{*}$. In particular, if the polytope $P$ is generated by facets of the form $\{\mathbf{y}\in P\:|\:\langle\mathbf{y},\mathbf{v}\rangle=-\alpha\}$ with $\alpha>0$, then its polar dual $P^*$ is generated by vertices of the form $\mathbf{v}/\alpha$.
\end{prop}

%
%
%\begin{lemma}
%Suppose that $P$ is a non-codegenerate polyhedron.  Then $\VolC(P)$ is well-defined.
%\end{lemma}
%\begin{proof}
%\end{proof}

\begin{lemma}\label{lem:same-function-interior}
If $P\subseteq\mathbb{R}^d$ is a full dimensional polytope and $\mathbf{0}$ is in its interior, then we have $\VolC(P)=\Vol(P^\pol).$
\end{lemma}


\Cref{def:dual-volume} is motivated by \Cref{lem:same-function-interior}, and intuitively, one views $\VolC$ as a volume function. To be precise, the notion of the dual volume $\VolC$ is much more powerful, in the sense that $\VolC(P)$ is always defined whenever $\mathbf{0}$ is not contained in the affine span of any facets of $P$, and the rational function $\VolC_{\z}(P)$ is always well-defined. %However, the polar dual $P^\pol$ is very often unbounded, including the situation when $P$ is a polytope that does not contain $\mathbf{0}$ in its interior. 


\begin{example}\label{ex: quadrilateral-dual-volume}
Consider a polytope $P\subset\R^2$ as the convex hull of $(1,1)$, $(2,1)$, $(3,-1)$, $(1,-1)$, with its normal fan $\N(P)$ shown in \Cref{fig:dual-volume-polytope-example-1}. We can pick $\mathbf{v_1}=(0,1)$, $\mathbf{v_2}=(1,0)$, $\mathbf{v_3}=(0,-1)$ and $\mathbf{v_4}=(-2,-1)$. Summing cyclically with $\mathbf{v_5}=\mathbf{v_1}$, \[f_{\N(P),\mathbf{v}}(u_1,u_2,u_3,u_4)=\sum_{i=1}^4\frac{|\det(\mathbf{v_i},\mathbf{v_{i+1}})|}{u_iu_{i+1}}=\frac{1}{u_1u_2}+\frac{1}{u_2u_3}+\frac{2}{u_3u_4}+\frac{2}{u_4u_1}.\]
We have $h_P(\mathbf{v_1})=1$, $h_P(\mathbf{v_2})=-1$, $h_P(\mathbf{v_3})=1$ and $h_P(\mathbf{v_4})=5$. Assigning $u_i=h_P(\mathbf{v_i})$ in $f_{\N(P),\mathbf{v}}$ for all $i$, we get  $\VolC(P)=(-1)+(-1)+\frac{2}{5}+\frac{2}{5}=-\frac{6}{5}$.
Assigning $u_i=h_{P-\z}(\mathbf{v_i})=h_P(\mathbf{v_i})+\langle \z,\mathbf{v_i}\rangle$ for all $i$, we have 
\begin{align*}\VolC_{\z}&(P)\\&=\frac{1}{(1{+}z_2)({-}1{+}z_1)}+\frac{1}{({-}1{+}z_1)(1{-}z_2)}+\frac{2}{(1{-}z_2)(5{-}2z_1{-}z_2)}+\frac{2}{(5{-}2z_1{-}z_2)(1{+}z_2)}.\end{align*}
%\lei{The $h_{P-\z}(\v)$ formula in this example used to be $h_{P-\z}(\v) = h_P(\z) - \langle \z, \v\rangle$. I have corrected it to $h_{P-\z}(\v) = h_P(\z) + \langle \z, \v\rangle$. }
\begin{figure}[ht!]
\centering
\begin{tikzpicture}[scale=0.8]
\draw[fill=black,opacity=0.2](1,1)--(2,1)--(3,-1)--(1,-1)--(1,1);
\draw[thick](1,1)--(2,1)--(3,-1)--(1,-1)--(1,1);
\node at (1,1) {$\bullet$};
\node at (2,1) {$\bullet$};
\node at (3,-1) {$\bullet$};
\node at (1,-1) {$\bullet$};
\node[above] at (1,1) {$(1,1)$};
\node[above] at (2.5,1) {$(2,1)$};
\node[below] at (3,-1) {$(3,-1)$};
\node[below] at (1,-1) {$(1,-1)$};
\draw[thin](0,-2)--(0,2);
\draw[thin](-1,0)--(4,0);
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[scale=0.8]
\draw[fill=black,opacity=0.2](0,1.6)--(1.6,0)--(0,-1.6)--(-1.6,-0.8);
\node at (0,0) {$\bullet$};
\draw[->,thin](0,0)--(0,2);
\draw[->,thin](0,0)--(0,-2);
\draw[->,thin](0,0)--(2,0);
\draw[->,thin](0,0)--(-2,-1);
\node[right] at (0,1.7) {$\mathbf{v_1}$};
\node[below] at (1.7,0) {$\mathbf{v_2}$};
\node[left] at (0,-1.7) {$\mathbf{v_3}$};
\node[above] at (-2,-1) {$\mathbf{v_4}$};
\end{tikzpicture}
\caption{A polytope $P$ and its normal fan $\N(P)$}
\label{fig:dual-volume-polytope-example-1}
\end{figure}
\end{example}
\begin{example}
Consider the unbounded polyhedron $P\subset\R^2$ defined by the inequalities $3x_1+x_2+3\geq0$, $x_1+x_2+1\geq0$, $-2x_1+x_2+4\geq0$. Its normal fan, $\N(P)$, is not a complete fan, shown in \Cref{fig:dual-volume-polyhedron-example-1}. We can pick $\mathbf{v_1}=(-2,1)$, $\mathbf{v_2}=(1,1)$ and $\mathbf{v_3}=(3,1)$. Compute that $h_P(\mathbf{v_1})=4$, $h_P(\mathbf{v_2})=1$, $h_P(\mathbf{v_3})=3$ and \[f_{\N(P),\mathbf{v}}(u_1,u_2,u_3)=\frac{3}{u_1u_2}+\frac{2}{u_2u_3}.\]
Substituting $u_i=h_P(\mathbf{v_i})$ for all $i$ gives $\VolC(P)$ and substituting $u_i=h_P(\mathbf{v_i})+\langle\z,\mathbf{v_i}\rangle$ gives $\VolC_\z(P)$. We therefore obtain $\VolC(P)=17/12$ and \[\VolC_{\z}(P)=\frac{3}{(4-2z_1+z_2)(1+z_1+z_2)}+\frac{2}{(1+z_1+z_2)(3+3z_1+z_2)}.\]
%\lei{Similar to above. }
\begin{figure}[ht!]
\centering
\begin{tikzpicture}[scale=0.4]
\draw[fill=black,opacity=0.2](-2,3)--(-1,0)--(1,-2)--(3.5,3);
\draw[thick](-2,3)--(-1,0)--(1,-2)--(3.5,3);
\draw[thin](-3,0)--(4,0);
\draw[thin](0,-3)--(0,4);
\node[below] at (-2,0) {$(-1,0)$};
\node[right] at (1,-2) {$(1,-2)$};
\node at (-1,0) {$\bullet$};
\node at (1,-2) {$\bullet$};
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[scale=0.4]
\draw[fill=black,opacity=0.2](0,0)--(-3.2,1.6)--(2.4,2.4)--(4.8,1.6);
\node at (0,0) {$\bullet$};
\draw[->,thin](0,0)--(-4,2);
\draw[->,thin](0,0)--(3,3);
\draw[->,thin](0,0)--(6,2);
\node[below] at (-4,2) {$\mathbf{v_1}$};
\node[left] at (3,3) {$\mathbf{v_2}$};
\node[below] at (6,2) {$\mathbf{v_3}$};
\node at (0,-3) {};
\node at (0,4) {};
\end{tikzpicture}
\caption{An unbounded polyhedron $P$ and its normal fan $\N(P)$}
\label{fig:dual-volume-polyhedron-example-1}
\end{figure}
\end{example}
\section{Properties of dual volume}\label{sec:properties}
\subsection{An integral formula}

\begin{theorem}\label{thm: dual vol is integral}
Let $P\subset\R^d$ be a polyhedron and $\z\in \R^d$ be any point such that $P-\z$ is non-codegenerate. Then \begin{align}\label{eqn: dual volume = exp integral}
\VolC_{\z}(P)=\int_{\R^d}\exp(-h_P(\v) - \langle\v,\z\rangle)d\v.
\end{align}
\end{theorem}

For a cone $C \subseteq\R^d$, define the \emph{dual cone} $C^{\dual}$ by
 \begin{equation}
 \label{eq:dualcone}
 C^\dual := \{\v\in \R^d\; | \; \langle \v,\y \rangle \geq 0\; \text{ for all } \y\in C \}. 
 \end{equation}
Note that we use ${}^*$ for polarity/duality on cones and ${}^\vee$ for polarity on polytopes.

\begin{cor}\label{cor:intform}
Let $P\subset\R^d$ be a non-codegenerate polyhedron. Then \begin{align}
\VolC(P)=\int_{\R^d}e^{-h_P(\v)}d\v.
\end{align}
\end{cor}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{comment}
\subsection{Canonical form and the valuative property}
In this subsection, we work briefly in the projective space $\mathbb{P}^d$. There is a natural embedding $\R^d\subset\mathbb{P}^d$ via $(x_1,\ldots,x_d)\mapsto[1:x_1:\cdots:x_d]$. An \emph{orientation} of a polytope is an orientation of its interior $\Int(P)$. We will use the language of canonical forms, developed by Arkani--Hamed, Bai and Lam \cite{ABL}. We also refer the readers to the survey \cite{lam2022invitation}.
\begin{theorem}\cite[Section 6]{ABL}\cite[Theorem 1]{lam2022invitation}
For each full dimensional polyhedron $P\subset\R^d\subset\mathbb{P}^d$, there exists a rational $d$-form $\Omega(P)$ on $\mathbb{P}^d$, with poles only along facet hyperplanes, and that these poles are simple, such that $\Omega(P)$ is uniquely determined recursively by the followings:
\begin{enumerate}
\item $\Omega(\mathrm{pt})=\pm1$ depending on orientations;
\item for any facet $F$, $\mathrm{Res}_{H}\Omega(P)=\Omega(F)$, where $H$ is the hyperplane spanned by $F$.
\end{enumerate}
\end{theorem}

The following result is essentially established in \cite[Section 7.4]{ABL}.
\begin{theorem}\label{thm:dual-volume-canonical-form}
Suppose $P\subset\R^d\subset\mathbb{P}^d$ is full dimensional, we have \[\Omega(P)(\x)=\VolC_{\x}d^d\x.\]
\end{theorem}
\begin{proof}
When $\x\in\Int(P)$, \cite[Section 7.4]{ABL} and \cite[Theorem 3]{lam2022invitation} say that $\Omega(P)(\x)=\Vol((P-\x)^*)d^d\x$, which equals $\VolC_{\x}d^d\x$ by \Cref{lem:same-function-interior}. Since $\VolC_{\x}$ is a well-defined rational function, by analytic continuation, we obtain the desired statement.
\end{proof}
\Yibo{Need some background on valuations. What's the best reference?}
\begin{theorem}\label{thm:dual-volume-is-valuation}
The dual volume functions $\VolC$ and $\VolC_{\z}$ are valuations on polyhedra.
\end{theorem}
\begin{proof}
The valuative property of $\VolC$ follows from that of $\VolC_{\z}$ by specializing $\z$ to $\mathbf{0}$. It suffices to show that for a full dimensional polyhedron $P$ and a decomposition $P=P_1\cup\cdots\cup P_n$ such that all $P_i\cap P_j$'s are lower dimensional, we have \[\VolC_{\z}(P)=\VolC_{\z}(P_1)+\cdots\VolC_{\z}(P_n)+\sum_{A\subset[n],|A|\geq2}c_A\VolC_{\z}\left(\bigcap_{a\in A}P_a\right)\]
if on the level of indicator functions, we have \[\mathds{1}_{P}=\mathds{1}_{P_1}+\cdots+\mathds{1}_{P_n}+\sum_{A\subset[n],|A|\geq2}c_A\mathds{1}_{\cap_{a\in A}P_a},\]
for some real numbers $c_A$'s. By definition, $\VolC_{\z}(Q)=0$ if $Q$ is not full dimensional. By \cite[(7.21)]{ABL}, $\Omega(P)=\Omega(P_1)+\cdots\Omega(P_n)$ and using \Cref{thm:dual-volume-canonical-form}, we are done.
\end{proof}
\end{comment}

%\section{The dual interior cone of polyhedra}

\subsection{Valuative property}\label{section: valuation}
%\TL{I will use $*$ for polarity on cones and $\vee$ for polarity on polytopes.}


Let $[P]$ be the indicator function of a polyhedron $P$.  Let $\Pcal$ denote the space spanned by indicator functions of polyhedra. We show that $\VolC(P)$ and $\VolC_\z(P)$ are valuative.  

%\TL{Mention work of Filliman and Kuperberg} \Lei{See remark below}


\begin{theorem}\label{thm:valuation}
Suppose $\sum_{i=1}^r \alpha_i [P_i] = 0$
in $\Pcal$, for polyhedra $P_1,\ldots,P_r\in\R^d$ and $\alpha_1,\ldots,\alpha_r \in \R$. Then
\begin{equation}\label{eq:valz}
\sum_{i=1}^r \alpha_i \Vol_\z^\vee(P_i) = 0.
\end{equation}
If each $P_i$ is non-codegenerate, then
\begin{equation}\label{eq:noncodeg}
\sum_{i=1}^r \alpha_i \Vol^\vee(P_i) = 0.
\end{equation}
\end{theorem}


 We deduce \Cref{thm:valuation} from the classical result of Lawrence stating that the algebra of indicator functions of cones has a polarity involution.  The result is also known to Alexander Barvinok \cite{Barvinokprivatecommunication}. It generalizes duality results of Filliman~\cite{filliman} and Kuperberg~\cite{Kuperburg}.




\iffalse
\begin{remark}\label{rmk: compare with filliman and kuperburg}
The case of \Cref{thm:valuation} where the $P_i$'s are all simplices containing the origin was established by Filliman~\cite{filliman} in 1990, and is known as Filliman's duality. In 2003, Kuperburg \cite{Kuperburg} extends Filliman's duality to an involution on the space of polytopal measures of non-codegenerate polytopes, and we have borrowed the terminology ``non-codegenerate'' from Kuperberg. Our results extends both results to include all (possibly unbounded) polyhedra; see also discussion at the end of \cite{Kuperburg}.  \Cref{thm:valuation} is also known to Alexander Barvinok~\cite{Barvinokprivatecommunication}. 
\end{remark}

We prove \Cref{thm:valuation} using the polytope algebra.

\fi

\subsection{Relation to adjoint polynomials}\label{sec:adjoint}

The adjoint polynomial of a polytope was first introduced by Warren in \cite{Warren}.  We use the version given in Aluffi \cite{Aluffi}; see also \cite{Kohn-Ranestad}.


\begin{defin}[{\cite[Definition/Theorem 4.1]{Aluffi}}]\label{def-adjoint}
Let $C$ be a polyhedral cone in $\R^{d+1}$ generated by the extreme rays $V(C)$, and let $T$ be a triangulation of $C$. The \emph{adjoint polynomial} of $C$ is given by
\begin{equation}\label{eqn: def-adjoint}
\adj_C(\z) = \sum_{F \in T} |\det(F)| \prod_{\v \in V(C) \setminus V(F)} \langle \v, \z\rangle
\end{equation}
where the sum is over all simplicial cones $F = \sp_{\R_{\geq 0}}(\v_1,\ldots,\v_{d+1})$ in $T$, and $\det(F) = \det(\v_1,\ldots,\v_{d+1})$.
\end{defin}
This definition is independent of the choice of the triangulation $T$.

Combine the summands in the definition of $\VolC_{\z}(P)$ into a single fraction to get:

\begin{align}\label{eqn: VolC one fraction}
\begin{split}
 \VolC_\z(P) &=\frac{A_{\z}(P)}{B_{\z}(P)},
 \end{split}
 \end{align}
where 
\begin{equation} \label{eq:denominator of VolC}
\begin{split}
A_{\z}(P) &:= \displaystyle\sum\limits_{C\in \mathcal{T}}|\det(C)| \displaystyle \prod\limits_{\v\in V(\N(P))\setminus V(C)} h_{P-\z}(\v),   \\
B_{\z}(P) &:=  \displaystyle\prod_{\v\in V(\N(P))} h_{P-\z}(\v)
\end{split}
 \end{equation}
 and $\mathcal{T}$ is any triangulation of $\N(P)$, $C$ is a simplicial cone in $\mathcal{T}$ generated by its extreme rays $V(C) = \{\mathbf{v}_{1},\ldots,\mathbf{v}_{d}\}$, and $\det(C) = \det(\v_1 , \dots, \v_d)$. 

 
%For the rest of the paper we will call $A_{\z}(P)$ (resp. $B_{\z}(P)$) the \emph{numerator} (resp. \emph{denominator}) of $\VolC_{\z}(P)$.


We show that the numerator of the dual volume function $\Vol_\z(P)$ coincides with the adjoint polynomial of (the cone of) the dual polytope $C(P)^*$.

%The following theorem shows that the numerator of $\Vol^\vee_\z(P)$ is the adjoint polynomial of the cone $C(P)^*$.
 
\begin{theorem}\label{thm:adjoint} For any vector $\z \in \R^d$, let $\bar{\z} = (1, \z) \in \R^{d+1}$. Then 
\begin{equation*} \VolC_{\z} (P) =\frac{\adj_{C(P)^*}(\bar{\z} ) }{B_{\z}(P)}, \qquad \text{or equivalently,} \qquad  \adj_{C(P)^*}(\bar{\z}) = A_{\z}(P).
\end{equation*}
\end{theorem}

\subsection{Canonical forms}\label{sec:canform}
\def\PP{{\mathbb{P}}}
We connect the dual volume of a polyhedron to positive geometry.  \emph{Positive geometries} are semialgebraic sets endowed with a distinguished meromorphic form called the \emph{canonical form}. \emph{Projective polytopes} $\bar P \subseteq \PP^d(\R)$ are examples of positive geometries (see 
\cite{ABL,lam2022invitation}). 

\begin{theorem}\label{thm:canformdualvol}
Let $P \subseteq \R^d$ be a full-dimensional polyhedron that does not contain lines.  Then the canonical form $\Omega(\bar P)$ of the projective polytope $\bar P$ is given by 
\[\Omega(\bar P) = \Vol^\vee_\z(P) dz_1 dz_2 \cdots dz_d.\]
\end{theorem}


\iffalse
Let $P \subseteq \R^d$ be a full-dimensional polyhedron not containing any lines.  View the projective space $\PP^d = \R^d \sqcup H_\infty$ as the union of the affine space and a plane at infinity.  Then the closure $\bar P \subseteq \PP^d(\R)$ is a \emph{projective polytope}.  If $P$ is bounded, that is, it is a polytope, then $\bar P = P$.  If $P$ is an unbounded polyhedron, then $\bar P$ is the union of $P$ together with the face $\bar P \cap H_\infty$ at infinity, which could be a vertex, a facet, or a face of intermediate dimension.

The following result states that (oriented) projective polytopes are positive geometries.
\begin{theorem}[{\cite[Section 6.1]{ABL}}]\label{thm:canformexist}
For each full-dimensional oriented projective polytope $\bar P \subseteq \PP^d(\R)$, there exists a unique nonzero meromorphic $d$-form $\Omega(\bar P)$ on $\PP^d$, called the \emph{canonical form}, satisfying the following recursive property.  If $d > 0$, the canonical form $\Omega(\bar P)$ has poles exactly along the facets of $\bar P$, all these poles are simple, and for each facet $F$, the residue of $\Omega$ along $F$ is given by
$$
\Res_F(\Omega(\bar P)) = \Omega(F).
$$
If $d = 0$, then $\bar P$ is a point and we have $\Omega = \pm 1$ depending on the orientation. 
\end{theorem}

\begin{remark}
    Since all the projective polytopes we consider in this paper arise as closures of polyhedra $P \subseteq \R^d$, they inherit a natural orientation from any chosen orientation of $\R^d$.  Thus in the following we will ignore the orientation.
\end{remark}


We shall use the following basic result.
\begin{thm}[\cite{ABL}]\label{thm:canformsub}
Suppose that $\{Q_1,\ldots,Q_n\}$ is a subdivision of a projective polytope $\bar P$.  Then \[\sum_{i=1}^n \Omega(Q_i) = \Omega(\bar P).\]
\end{thm}

\begin{remark}
    An alternative proof is to directly compare \eqref{eq:volsimplex} with the canonical form of a projective simplex, given in \cite[(5.7)]{ABL}.
\end{remark}





\begin{remark}
Note that if $P \subseteq \R^d$ contains a line then the closure $\bar P \subseteq \Pbb^d(\R)$ is not a projective polytope.  Nevertheless, it can be considered a pseudo-positive geometry \cite{ABL}, and has canonical form $0$, agreeing with \Cref{lem:Vol0}.
\end{remark}

Combining \Cref{thm:canformdualvol} with \Cref{thm:adjoint}, we obtain the following; see \cite{Gaetz} for an alternative approach.
\begin{cor}\label{cor:adjointzero}
Let $P \subsetneq \R^d$ be a full-dimensional polyhedron not containing lines.  Then $\Omega(\bar P) = \frac{\adj_{C(P)^*}(\bar \z)}{B_\z(P)} dz_1 dz_2 \cdots dz_d$.  Thus the zero set of $\Omega(\bar P)$ is the vanishing set of $\adj_{C(P)^*}$, known as the \emph{adjoint hypersurface} of $P^\vee$.
\end{cor}

\def\PGL{{\rm PGL}}
The definition of the canonical form $\Omega(\bar P)$ for a projective polytope is invariant under automorphisms of $\PP^d$.  In particular, if $\bar P$ and $\bar P'$ are related by a projective transformation $g \in \PGL(d+1)$, we have $\Omega(\bar P) = g^* \Omega(\bar P')$.  This leads to the following (perhaps unexpected) property of the dual volume function.

\begin{cor}\label{cor:proj}
    Let $P \subseteq \R^d$ and let $P' = g \cdot P$ where $g \in \PGL(d+1)$ is a projective transformation.  Let $\z' = (z'_1,\ldots,z'_d)$ be the image of the coordinates $z_1,\ldots,z_d$ under $g$.  Then 
    $$
    \Vol_\z(P) =  \det\left(\frac{\partial z'_j}{\partial z_i}\right) \Vol_{\z'}(P').
    $$
\end{cor}


\begin{example}
    Suppose that $P = [-1,2]$ and $P' = [-2,4]$.  Then we have $z' = 2z$.  So
    $$
    \Vol_z([-1,2]) = 2 \Vol_{2z}([-2,4])  = \frac{1}{4-(2z)} - \frac{1}{-2-(2z)} = \frac{1}{2-z} - \frac{1}{-1-z}.
    $$
\end{example}

\begin{example}
Let $P \subseteq \R^2$ be the triangle with vertices $(0,3),(0,1),(1,1)$ and $P'$ be the triangle with vertices $(0,3/2),(0,1/2),(1,1/2)$.  The projective transformation $(z_1,z_2)\mapsto (z'_1,z'_2) = (z_1/(z_2+1),z_2/(z_2+1))$ takes $P$ to $P'$.  We have
$$
\VolC_\z(P)  =\frac{1}{(1+z_2)^3} \frac{1}{z'_1(z'_2 - \frac{1}{2})(\frac{3}{2} - z'_1 - z'_2)}= \frac{4}{z_1 (3 - 2 z_1 + z_2) ( z_2-1)}.
$$
\end{example}

\begin{comment}
\begin{example}
    Suppose that $P = [a,b]$ and $P' = [1/a,1/b]$, where $z' = g(z) = 1/z$.  So
    \begin{align*}
    \Vol_z([a,b]) &= \frac{1}{z^2} \Vol_{1/z}([1/a,1/b])  \\
    &=(-\frac{1}{z^2}) (\frac{1}{1/b-1/z} - \frac{1}{1/b-1/z}) \\
    &= \frac{1/b-1/a}{(z/b - 1)(z/a-1)} = \frac{a-b}{(b-z)(a-z)} .
    \end{align*}
\end{example}
\end{comment}



\fi
\section{Dual mixed volumes}\label{sec:Minkowski}
%\TL{Change $\x$ to $\blambda$?}



\begin{comment}
For convex bodies $P,Q \subseteq \R^d$, the \emph{Minkowski sum} is defined by
$$
P+Q := \{\p+\q \;|\; \p \in P, \q\in Q\}.
$$
More generally, for closed convex bodies $S_1,\ldots,S_r \subseteq \R^d$ and positive real numbers $x_1,\ldots,x_r$, we define
$$
\x \S = x_1 S_1 + \cdots + x_r S_r := \left\{\p_1 + \p_2 + \cdots + \p_r \;|\; \frac{1}{x_i}\p_i \in S_i \text{ for all } i\right\}.
$$

Recall in \eqref{eq:support} the support function $h_S$ of $S$ is defined for, and uniquely determines, any closed convex body $S$.  It follows directly from the definition that, the support function of the Minkowski sum $\x\S$ is given by 
\begin{equation}\label{eq:hxS}
h_{\x\S} = x_1 h_{S_1} + x_2 h_{S_2} + \cdots + x_r h_{S_r}.
\end{equation}
We use this formula to extend the definition of $\x \S$ to arbitrary real parameters $x_1,\ldots,x_r \in \R$.  Namely, $\x\S$ is the ``object'' with support function $h_{\x\S}$ given by \eqref{eq:hxS}.  It is an interesting question to characterize the parameters $\x$ such that $h_{\x\S}$ is the support function of a closed, convex set.  Note that the support functions of closed, bounded, convex sets are exactly the positive homogeneous, convex, real-valued functions on $\R^d$.

Suppose that $h_{\x\S} = h_{T}$ for some closed convex set $T$. Furthermore, suppose that $x_1,\ldots,x_s \geq 0$ and $x_{s+1},\ldots,x_r \leq 0$.  It follows then that we have 
$$
x_1 h_{S_1} + \cdots + x_s h_{S_s} = h_T + (-x_{s+1}) h_{S_{s+1}} + \cdots + (-x_r) h_{S_r}.
$$
Equivalently, $T + \sum_{i=s+1}^r (-x_i) S_i = \sum_{j=1}^s x_j S_j$, where we are taking usual Minkowski sums of convex bodies.  Thus our definition of $\x\S$ is consistent with Minkowski sums when $\x\S = T$ is a closed convex set.
\end{comment}

We move on to consider a sequence $\P = (P_1,\ldots,P_r)$ of polyhedra in $\R^d$.  Each $h_{P_i}$ is piecewise-linear, and their common domains of linearity give a fan $\F$ in $\R^d$, which coincides with the normal fan $\N(P)$ of the Minkowski sum $P = P_1+ \cdots + P_r$.  The fan is complete exactly when $P$ (and thus each $P_i$) is a polytope.  In the following  assume that the Minkowski sum $P$ is full-dimensional in $\R^d$, so the maximal cones of $\F$ are pointed.


\begin{comment}
The following lemma is straightforward.
\begin{lemma}\label{lem:hxP}
For any $x_i \in \R$, the piecewise-linear function $h_{\x\P}$ restricts to a linear function on any of the maximal cones of $\F$.  In particular, the function $h_{\x\P}$ is uniquely determined by its values on the rays of $\F$.
\end{lemma}

Call $\P$ \emph{regular} if $\P$ satisfies \eqref{eq:full} and there exists a point $\x \in \R^r$ so that $h_{\x\P}(\v) \neq 0$ for all rays $\R_{\geq 0} \cdot \v$ of $\F$.  Not all sequences $\P$ of polytopes are regular, even when \eqref{eq:full} is satisfied.  For example if $\P = (P)$ is a single full-dimensional polytope, then $\P$ is regular if and only if $P$ is non-codegenerate. 

\begin{proposition}\label{prop:reg}
The sequence $\P = (P_1,\ldots,P_r)$ is regular if and only if for each ray $\R_{\geq 0} \cdot \v$ of $\F$, there exists a $j$ such that $h_{P_j}(\v) \neq 0$.
\end{proposition}
Since for any ray $\R_{\geq 0}\cdot \v$ there exists some standard unit vector $P= \e_i$ such that $h_P(\v) = -\langle \v , \e_i\rangle \neq 0$. The following proposition follows from \Cref{prop:reg}.

\begin{proposition}\label{prop:mpxz-regular}
Suppose that \eqref{eq:full} is satisfied and $\P = (P_1,P_2,\ldots,P_r, -\e_1,-\e_2,\ldots,-\e_d)$.  Then $\P$ is regular.
\end{proposition}
\end{comment}

\begin{defin}\label{def:dual-mixed-volume}
Let $\P = (P_1,\ldots,P_r)$ be a regular sequence of polyhedra with normal fan $\N(P)$ and let $\x = (x_1, \dots, x_r)$. The \emph{dual mixed volume rational function} $m_\P(\x)$ is
$$
m_\P(\x) := \VolC(h_{\x\P}) = f_{\N(P),\mathbf{v}}(h_{\x\P}(\mathbf{v_1}),\ldots,h_{\x\P}(\mathbf{v_g}))
$$
where $\v_1, \dots, \v_g$ are the generating rays of $\mathcal{N}(P)$, with notation as in \Cref{def:dual-volume}.  For any sequence $\P$ that is full-dimensional, also define
$$
m_\P(\x,\z) := m_{(P_1,P_2,\ldots,P_r,-\e_1,\ldots,-\e_r)}(x_1,\ldots,x_r,z_1,\ldots,z_d) = \VolC(h_{\x\P-\z}).
$$
\end{defin}
If $\P$ is not full-dimensional,  we set $m_\P(\x) := 0$ and $m_{\P}(\x,\z):=0$. By \Cref{def:universal-dual-volume} we can write $m_{\P}$ as a rational function in $x_1, x_2, \dots, x_r$, with degree $-d$. The denominator is a product of linear factors, each corresponding to a ray $\v_i$ of $\mathcal{N}(P)$.  
Note that $m_\P(\x)$ generalizes both $\VolC$ and $\VolC_\z$.  For a full-dimensional polytope $P$, we have the specializations
$$
m_P(1) = \VolC(P), \qquad m_{(P,-\e_1,-\e_2,\ldots,-\e_d)}(1,z_1,z_2,\ldots,z_d) = \VolC_\z(P).
$$
\begin{comment}
\subsection{Mixed polar sets}

We next consider the mixed analogue of \Cref{lem:same-function-interior}. For the polar dual polyhedron $(\x\P)^\pol$ to exist, we need to have the origin inside the interior of $\x\P$. The support function $h_{\x\P}$ can help us characterize such a condition.

\begin{proposition}\label{prop: support function and containing origin}
    Let $P \subseteq \R^d$ be a polyhedron.  Then $h_P \geq 0$ if and only if $\0 \in P$, and $h_P > 0$ on $\R^d \setminus \0$ if and only if $\0$ is  in the interior of $P$.
\end{proposition}


From now on, we assume that $\P$ is a regular sequence of polyhedra. We define the following polyhedral cones in $\R^r$:
\begin{align*}
A_\P &:= \{ \x \mid h_{\x\P} \geq 0 \text{ on } \R^d\},\text{ and} \\
A_\P^\circ &= \{\x \mid h_{\x\P} > 0 \text{ on } \R^d\setminus \0\}.
\end{align*} 
By \Cref{lem:hxP}, $\x \in \P$ if and only if $h_{\x\P}(\v_i) \geq 0$ for each ray $\R_{\geq 0} \cdot \v_i$ of the fan $\F = \mathcal{N}(P)$.  Thus the cone $A_\P$ is a nonempty closed polyhedral cone, and $A_\P^\circ$ is an open (possibly empty) polyhedral cone, the interior of $A_\P$.  By \Cref{prop: support function and containing origin} the cones $A_\P$ and $A_\P^\circ$ correspond to polyhedra $\x\P$ that contain the origin and those that contain the origin in the interior, respectively.

When $\x\in A_\P^\circ$, we define the polar dual of $\x\P$ to be the set 
$$
(\x\P)^\pol:= \{\w \in \R^d \mid h_{\x\P}(\w) \leq 1\} \subseteq \R^d,
$$ which is clearly closed and bounded, and contains the origin.  When $\x\P$ is a polytope, then this definition agrees with the usual polar of $\x\P$.
\begin{proposition}
Suppose that $\x \in A_\P^\circ$.  Then 
$$
m_\P(\x) = \Vol((\x\P)^\pol).
$$
\end{proposition}

\end{comment}


\begin{theorem}
Let $\P = (P_1, \dots, P_r ) $ be a sequence of polyhedra in $\R^d$, $\x = (x_1, \dots, x_r)$ in $\R^r$, and $\z \in \R^d$ be any point such that $\x\P - \z$ is non-codegenerate. Then
\begin{align*}
 m_\P(\x,\z) = \int_{\R^d} \exp(-h_{\x\P-\z}(\v)) d\v.
\end{align*}
\end{theorem}

If $\x\P$ is non-codegenerate, we can let $\z \to \0$ and take limits on both sides to get:
\begin{align*}
 m_\P(\x) = \int_{\R^d} \exp(-h_{\x\P}(\v)) d\v.
\end{align*}

%\section{Fine mixed cells}\label{sec:fine}

We study the behavior of the dual mixed volume function $m_\P(\x)$ under \emph{mixed subdivisions}: those subdivision of $P_1+\cdots+P_r$ that are compatible with the Minkowski sum.  Our main result (\Cref{thm: mixed subdivision formula}) states that dual mixed volume is additive under mixed subdivisions, which generalizes \Cref{thm:valuation}  to the mixed setting.  


\begin{theorem}\label{thm: mixed subdivision formula}
Let $\mathcal{S} = \{\Q^{(1)} , \Q^{(2)}, \dots, \Q^{(n)}\}$ be a mixed subdivision of $\P = (P_1, \dots, P_r)$, and let $\x = (x_1, x_2, \dots, x_r)$. Then the dual mixed volume of $\P$ can be written as a sums of the dual mixed volumes of the cells in  $\mathcal{S}$:
\begin{equation}\label{eqn: mixed subdivision formula}
m_{\P} (\mathbf{x},\z) = \sum_{ \Q \in \mathcal{S} } m_{\Q} (\mathbf{x},\z).
\end{equation}
\end{theorem}

\begin{comment}
To prove \Cref{thm: mixed subdivision formula} we use the following known result; see for example \cite{HRS}. 

\begin{lemma}\label{lem:xS}
Suppose that $\x = (x_1, x_2, \dots\, x_r)$ satisfies $x_i >0$ for all $i\in [r]$.  Then $\x\mathcal{S} = \{\x \Q^{(1)} , \x \Q^{(2)}, \dots , \x \Q^{(n)}  \}$ is a subdivision of $\x\P$. 
\end{lemma}
\end{comment}

%

%\begin{cor}
%Suppose that $\{Q_1,Q_2,\ldots,Q_n\}$ is a subdivision of a polytope $P$.  Then $\VolC(P) = \sum_{i=1}^n \VolC(Q_i)$ and 
%\end{cor}

\section{Formulae from the Cayley polytope}\label{sec:Cayley}

 \begin{defin}
 For a sequence of polytopes $\P = (P_1, P_2, \dots, P_r)$ in $\R^d$, let $\mu_i(\z) := (\z, \mathbf{e}_i)$ for $i \in [r]$, where $\mathbf{e}_i$ is the $i$-th standard basis vector in $\R^r$. The \emph{Cayley embedding} of $\P$ is the map $\mathcal{C}$ sending $\P$ to following polytope in $\R^{d+r}$.
\[ 
\mathcal{C}(\P):=\mathcal{C}(P_1, P_2, \dots, P_r) := \conv \{ \mu_1(P_1) \cup \mu_2(P_2) \cup \dots \cup \mu_r(P_r) \}.
\]

We call $\mathcal{C}(P_1, P_2, \dots, P_r)$ the \emph{Cayley polytope} of $(P_1, P_2, \dots, P_r)$.

\end{defin}

The following proposition is known as the Cayley trick.
\begin{prop}[{\cite[Theorem 3.1]{HRS}}]\label{prop: Cayley trick}
There is a bijection between the mixed subdivisions of $P_1 + P_2 + \dots + P_r$ and the subdivisions of the Cayley embedding $\mathcal{C}(P_1, P_2, \dots, P_r)$. Furthermore, it restricts to a bijection between the fine mixed subdivisions of $P_1 + P_2 + \dots + P_r$ and the triangulations of $\mathcal{C}(P_1, P_2, \dots, P_r)$. %The bijection is given by applying the Cayley embedding on each Minkowski cell in a mixed subdivision $\mathcal{S}$ of $P= P_1 + P_2 + \dots + P_r$. The Cayley polytopes of these cells form a subdivision of $\mathcal{C}(P_1, P_2, \dots, P_r)$.
\end{prop}

The Cayley trick allows us to write the dual mixed volume function as a single dual volume function calculated in an affine hyperplane, where volumes are normalized by the normal vector $\mathbf{u}=(0,\ldots,0,1,\ldots,1)$.
\begin{theorem}\label{thm:Cayleysub}
Let $\P=(P_1,\ldots,P_r)$ be full-dimensional. For $t=\langle\one,\x\rangle$, \[m_{\P}(\x,\z)=\frac{\prod_{i=1}^n x_i}{\sum_{i=1}^n x_i}\cdot\VolC_{(\z,\x)}(t\mathcal{C}(\P))\]
where the right hand side is calculated in the affine hyperplane $\{(\z,\x)\:|\: \langle\one,\x\rangle=t\}\subset\R^{d+r}.$
\end{theorem}


\section{Polytopes in an affine hyperplane}\label{sec:affine}

Many polytopes of interest in combinatorics, such as matroid polytopes, alcoved polytopes, and generalized permutohedra, are naturally defined to live in an affine hyperplane. We develop the following formalism to study this case.





%The following \emph{hyperplane dual volume function} will play the role of the \emph{dual volume function} for full dimensional polytopes that live inside the affine hyperplane $H_k$, including all important polytopes that appear in later sections.
\begin{defin}\label{def:hyperplane-dual-volume}
let $P$ be a polytope whose affine span is the hyperplane $H \subseteq \R^d$. Fix a triangulation $\T$ of the \textit{boundary} of $\Cone(P)^*$, and let $\{\v_i\}_{i=1}^g$ be generators for the extremal rays in $\T$. Let $F \in \T$ denote a top-dimensional ($(d-1)$-dimensional) simplicial cone $F$ in $\T$. The \emph{hyperplane dual volume function} of the ``affine polytope'' $P$ is 
\begin{equation}\label{eq:affinedual}
\EVolC_\z(P) := \sum_{F=\mathrm{span}(\v_{i_1},\ldots,\v_{i_{d-1}})\in\T}\frac{|\det(\v_{i_1},
\v_{i_2},\ldots,\v_{i_{d-1}},\one)|}{\langle \v_{i_1},\z \rangle \langle \v_{i_2},\z \rangle \cdots \langle \v_{i_{d-1}},\z \rangle}.
\end{equation}
If $P$ does not have full dimension in $H$, i.e. $\dim(P)<d-1$, we define $\EVolC_\z(P) :=  0$.
\end{defin}
%We reiterate that $\T$ is a triangulation of the boundary of $\Cone(P)^*$, not of $\Cone(P)^*$ itself. Also note that $\EVolC_\z(P)$ depends only on $\Cone(P)$; it is unchanged under replacing $P$ by a scalar multiple $tP$. 
Similar to the case of the dual volume (\Cref{def:dual-volume}), $\EVolC_{\z}(P)$ does not depend on the triangulation $\T$, and the following valuative property holds (similar to \Cref{thm:valuation}).
\begin{comment}
\begin{proposition}\label{prop:enrich}
The hyperplane dual volume function $\EVolC_\z(P)$ is a rational function of degree $1-d$ that does not depend on the choice of $\T$. Moreover, we can realize the hyperplane dual volume function as a dual volume function in $H_t$ with $\z_0\in\R^d$ as the origin (see \Cref{sub:normalization-affine} for the identification) where $t:=\langle\one,\z_0\rangle$. Then
\begin{equation}\label{eq:hyperplane-function-is-actual-dual-function}
\EVolC_{\z}(P)=\Vol^\vee_\z((t/k)P) \mbox{ for $\z \in H_t$},
\end{equation}
and
if the dual volume $\Vol^\vee((t/k)P)$ is well-defined, that is, $\z_0$ is not in the affine span of the facets of $(t/k)P$, we have
\begin{equation}\label{eq:hyperplane-volume-is-actual-dual-volume}
\EVolC_{\z}(P)\mid_{\z=\z_0}=\VolC((t/k)P).
\end{equation}
\end{proposition}




\begin{example}
Let $P$ be the simplex that is the convex hull of $(1,0,0),(0,1,0),(0,0,1) \in \R^3$, which lies in $H_1$. So the cone $\Cone(P)$ is the nonnegative octant of $\R^3$ and we can identify the dual cone $\Cone(P)^*$ as the nonnegative octant of $\R^3$ as well. Take rays $\v_1=(1,0,0)$, $\v_2=(0,1,0)$, $\v_3=(0,0,1)$.  The facets of $\Cone(P)^*$ are \[\sp_{\geq 0}(r_1,r_2), \qquad \sp_{\geq 0}(r_1,r_3), \qquad \sp_{\geq 0}(r_2,r_3),\]
each of which is a simplicial cone of dimension $2$ already. Then \[\EVolC_{\z}(P)=\frac{|\det(\v_1,\v_2,\one)|}{\langle\v_1,\z\rangle\langle\v_2,\z\rangle}+\frac{|\det(\v_1,\v_3,\one)|}{\langle\v_1,\z\rangle\langle\v_3,\z\rangle}+\frac{|\det(\v_2,\v_3,\one)|}{\langle\v_2,\z\rangle\langle\v_3,\z\rangle}=\frac{1}{z_1z_2}+\frac{1}{z_1z_3}+\frac{1}{z_2z_3}.\]
Now let us suppose that we take $\z_0 = (0,0,1) = \v_3$.  Then $H_1$ can be identified with $\R^2$ by the projection $(y_1,y_2,y_3) \mapsto (y_1,y_2)$, under which $P$ is identified with the convex hull of $(0,0), (1,0),(0,1)$ in $\R^2$.  Then one can calculate
\[\Vol^\vee_\y(P) = \frac{1}{y_1y_2} + \frac{1}{y_1(1-y_1-y_2)} + \frac{1}{y_2(1-y_1-y_2)},\]
which is equal to $\EVolC_\z(P)$ under the substitution $(z_1,z_2,z_3) = (y_1,y_2,1-y_1-y_2)$ for a typical point in $H_1$.
\end{example}
\end{comment}

\begin{thm}\label{thm:affinevaluation}
Suppose $
\sum_{i=1}^r \alpha_i [P_i] = 0
$
for polytopes $P_1,\ldots,P_r\in H$ and $\alpha_1,\ldots,\alpha_r\in\R$, then
$$
\sum_{i=1}^r \alpha_i \EVolC_\z(P_i) = 0.
$$
\end{thm}

\begin{defin}\label{def:hyperplane-dual-mixed}
The \emph{hyperplane dual mixed volume function} of $\P=(P_1,\ldots,P_r)$ is
\begin{equation}\label{eq:affinedualmixed}
\tilde m_{\P}(\x,\z) := \sum_{F=\mathrm{span}(\v_{i_1},\ldots,\v_{i_{d-1}})\in\T}\frac{|\det(\v_{i_1},\v_{i_2},\ldots,\v_{i_{d-1}},\one)|}{h_{\x\P-\z}(\v_{i_1})h_{\x\P-\z}(\v_{i_2}) \cdots h_{\x\P-\z}(\v_{i_{d-1}})},
\end{equation}
a rational function in $\x=(x_1,\ldots,x_r)$ and $\z=(z_1,\ldots,z_d)$, with $h_{\x\P}(\v)$ defined as before.
\end{defin}
\begin{comment}
\subsection{Fine mixed cells in an affine hyperplane}
Our setup in this section resembles that of \Cref{sec:fine}, but there are some differences. We will highlight the distinction.
\begin{defin}
A \emph{fine mixed cell in the affine hyperplane $H_1$} is a sequence $\P=(P_1,\ldots,P_r)$ of simplices in $H_1$ satisfying both of the conditions
\begin{itemize}
\item $\dim(P_1)+\cdots+\dim(P_r)=d-1$, and
\item $\dim(P_1+P_2+\cdots+P_r)=d-1$.
\end{itemize}
\end{defin}
Let $\P$ be a fine mixed cell and write $d_i=\dim(P_i)$ for $i=1,\ldots,r$ as before, and write $P=P_1+\cdots+P_r$. We describe $\Cone(P)^*$ as follows. Label vertices of $P_i$ as $\p_{i,1},\ldots,\p_{i,d_i+1}$ and for each $P_i$ with $d_i>0$, let $\v_{i,1},\ldots,\v_{i,d_{i}+1}$ be rays in $\R^d$ to be normal to $\mathrm{span}(P_j)$ for each $j\neq i$, and for each $a\in[d_i+1]$, 
\begin{align*}\label{eq:affine-fine-mixed-cell-rays}
&\max_{\y\in P_i}\langle\y,\v_{i,a}\rangle=\langle\p_{i,a},\v_{i,a}\rangle=1-\sum_{j\neq i}\langle \z_j,\v_{i,a}\rangle,\\
&\min_{\y\in P_i}\langle\y,\v_{i,a}\rangle=\langle\p_{i,b},\v_{i,a}\rangle=-\sum_{j\neq i}\langle\z_j,\v_{i,a}\rangle,
\end{align*}
for any $\z_j\in P_j$ and any $b\neq a$. The first equality signs determine the direction of $\v_{i,a}$ and the second equality sign tells us how to scale it. By counting dimensions, we see that $\v_{i,a}$'s are uniquely defined. As before, if $d_i=\dim(P_i)=0$, we just assume $\v_{i,1}=\frac{1}{r}\one$. This vector will not appear in any meaningful calculation. We do this scaling to ensure that $\langle\v_{i,1},\y\rangle=1$ for all $\y\in P=P_1+\cdots+P_r$. The idea of how $\v_{i,a}$'s are scaled is the same as in \Cref{sec:fine}, but the technical details are different.
\begin{lemma}\label{lem:affine-fine-mixed-cell-normal-fan}
With notations as above, $\sum_{a=1}^{d_i+1}\v_{i,a}=\frac{1}{r}\one$. Moreover, the boundary of $\Cone(P)^*$ has maximal cones $\{C_{\mathbf{a}}\:|\:\mathbf{a}=(a_1,\ldots,a_r)\in[d_1{+}1]\times\cdots\times[d_r{+}1]\}$ where \[C_{\mathbf{a}}=C_{a_1,\ldots,a_r}=\mathrm{span}_{\geq0}\bigcup_{i=1}^r\{\v_{i,a}\:|\:a\neq a_i\}.\]
\end{lemma}
As before, we define $\kappa(\P)$ to be the absolute value of the determinant with $d$ vectors $\v_{i,a}$'s together with $\one$, where $i\in[r]$ and $a\in A_i$ for any subset $A_i\subsetneq [d_i+1]$ with cardinality $d_i$. By \Cref{lem:affine-fine-mixed-cell-normal-fan}, as $\v_{i,1}+\cdots+\v_{i,d_i+1}$ is a multiple of $\one$, $\kappa(\P)$ is well-defined.
\begin{proposition}\label{prop:{lemma}affine-fine-mixed-cell-dual-volume}
Let $\P=(P_1,\ldots,P_r)$ be a fine mixed cell in $H_1$ and let $\v_{i,a}$ be as above. Then its hyperplane dual volume function is \[\tilde m_{\P}(\x,\z)=\kappa(\P)\prod_{i:d_i>0}\frac{x_i-\langle\x,\one\rangle/r+\langle\z,\one\rangle/r}{\prod_{a=1}^{d_i+1}(h_{\x\P}(\v_{i,a})+\langle\v_{i,a},\z\rangle)}\]
In particular, when $\langle\x,\one\rangle=\langle\z,\one\rangle$, this formula simplifies to \[\tilde m_{\P}(\x,\z)|_{\langle\x,\one\rangle=\langle\z,\one\rangle}=\kappa(\P)\frac{\prod_{i:d_i>0} x_i}{\prod_{i:d_i>0}\prod_{a=1}^{d_i+1}(h_{\x\P}(\v_{i,a})+\langle\v_{i,a},\z\rangle)}\]
\end{proposition}


\subsection{Fine mixed subdivisions in an affine hyperplane}
\end{comment}
% \Yibo{Need a definition for fine mixed subdivisions in $H_1$. I will work on this once \Cref{sec:mixedsub} is complete.}
From \Cref{thm: mixed subdivision formula} we obtain the following proposition.
\begin{prop}\label{prop:affine-fine-mixed-subdivision}
Let $\P=(P_1,\ldots,P_r)$ be a sequence of polytopes in $H_1$ such that $P=P_1+\cdots+P_r$ is full-dimensional in $H_r$. Let $\mathcal{S}=\{\Q^{(1)},\ldots,\Q^{(N)}\}$ be a fine mixed subdivision of $\P$. Then \[\tilde m_{\P}(\x,\z)|_{\langle\x,\one\rangle=\langle\z,\one\rangle}=\sum_{\Q\in\mathcal{S}}\tilde m_{\Q}(\x,\z)|_{\langle\x,\one\rangle=\langle\z,\one\rangle}.\]
\end{prop}



\section{Zonotopes} \label{sec:zonotope}
We study \emph{zonotopes}, polytopes that are Minkowski sums of intervals and show that their dual mixed volumes uniquely determined by dual mixed volumes of the deletion and contraction.  



%In this section, consider a sequence of intervals $\P=(P_1,\ldots,P_r)$ where $P_i=[-\p_i,\p_i]$ for $i=1,\ldots,r$. Assume $\P$ is full-dimensional, i.e. $\p_1,\ldots,\p_r$ span $\R^d$. Also assume for simplicity that $\p_i\neq0$ for all $i$ and $\p_i\neq\pm\p_j$ for $i\neq j$. Fine mixed subdivisions of $P_1+\cdots+P_r$ are called \emph{zonotopal tilings}.

%\subsection{Deletion-contraction formalism}
%We will use deletion-contraction to compute the dual mixed volume function of zonotopes. We set up the inductive steps in this section. 

Let $P \subseteq \R^d$ be a full-dimensional polytope and $\p \in \R^d$ be a vector.  Let $H$ be the hyperplane normal to $\p$ and let $H_{>0}$ (resp. $H_{\geq 0}$) and $H_{<0}$ (resp. $H_{\leq 0}$) denote the corresponding closed (resp. open) halfspaces. Let 
$$
P(x) := P+x[-\p,\p]
$$
The aim is to give a recursive description of the rational function
$$
V(x,\z):=\VolC_\z(P(x)) = \VolC_\z(P+x[-\p,\p]).
$$
Let $V(\z) = V(0,\z) = \VolC_\z(P)$.
Let $P_{/\p}:={\rm proj}_\p P$ denote the orthogonal projection of $P$ into $H$, in the direction of $\p$.
%The goal is to describe $V(x,\z)$ in terms of $V(\z)$, the ``deletion'' part, and $\VolC_{\z_0}(P_{/\p})$, the ``contraction'' part.  Here, $\z_0$ denotes a point in the hyperplane $H \subseteq \R^d$.

%Let the denominator $B_{\z}$ of $V(\z)$ as defined in \eqref{eq:denominator of VolC} equal $B_+(\z)B_0(\z)B_-(\z)$ where the three factors correspond to the rays of the normal fan of $P$ that belong to $H_{>0}$, $H$, and $H_{<0}$ respectively. We show that the three polynomials $B_+,B_0,B_-$ are pairwise relatively prime.

Let $B({\z})$ be the denominator of $V(\z)$ as defined in \eqref{eq:denominator of VolC}. The cones of the normal fan $\N'$ of $P+[-\p,\p]$ are obtained from $\N(P)$ by intersecting with the cones $H_{\geq 0}$, $H$, and $H_{\leq 0}$. Let $D(\z)$ be the product of linear factors $h_{P-\z}(\v_i)$'s where $\v_i$'s are rays of $\N'$ that are not rays of $\N$.

\begin{theorem}\label{thm:project}
There exists a unique pair of rational functions $W_+=W_+(\z)$ and $W_- = W_-(\z)$ satisfying the following properties:

\begin{itemize}
\item[(1).]  $W_{+}(\z) + W_{-}(\z) = V(\z)$;
\item[(2).]  $W_{\pm}(\z) = A_{\pm}(\z)/\big(B(\z)D(\z)\big)$ for some polynomial $A_{\pm}(\z)$;
\item[(3).]  writing $\z = \z_0 + t \p$ with $\z_0 \in H$, then $\lim_{t \to \infty} t W_{\pm}(\z_0,t) = \pm \Vol^\vee_{\z_0}(P_{/\p})/||\p||^2$.
\end{itemize}

Consequently, we have $V(x,\z) = W_+(\z+x\p)+W_-(\z-x\p)$.
\end{theorem}


Recursively applying \Cref{thm:project}, one can compute the dual mixed volume of a zonotope, via contraction-deletion.


By the Bohne--Dress Theorem, mixed subdivisions of zonotopes, or zonotopal tilings, are in bijection with one-element liftings of the corresponding oriented matroid.  Thus we obtain a formula for the dual mixed volume $m_\P(\x)$ for each such lifting.

\begin{comment}
\section{Generalized permutohedra}\label{sec:genperm}

We consider \emph{generalized permutohedra}, polytopes obtained by parallel translating the facet hyperplanes of a permutohedron. Details are omitted due to the length constraint.

\begin{theorem} We give an explicit formula for the (hyperplane) dual mixed volume of a generalized permutohedron.  
\end{theorem}

The combinatorics of fine mixed subdivisions of generalized permutohedra has been studied in \cite{postnikov-permutohedra}.  Our results lead to some curious identities that appear to be new.

Readers are referred to \cite{postnikov-permutohedra} for details on generalized permutohedra, which are a larger family of polytopes than zonotopes that we have considered, and appear everywhere in algebra and combinatorics.
% Not every generalized permutohedron is a zonotope.  To obtain the dual mixed volume of a more general family of generalized permutohedra, we consider Minkowski sums of simplices.
In this section, we switch our usual notation for dimension from $d$ to $n$ to be more consistent with the literature.

%In the literature, generalized permutohedra are defined to live in an affine hyperplane. 

Recall that $H_t = \{\y = (y_1,\ldots,y_n) \mid y_1+y_2 + \cdots + y_n = t\} \subseteq \R^n$.
\begin{defin}\label{def:generalized-permuohedra}
For $T\neq\emptyset\subset[n]$, define the following $(|T|-1)$-dimensional simplex \[\Delta_T := \conv(e_i \mid i \in T)\subseteq H_1.\]
Let $\P=(\Delta_T)_{T\in 2^{[n]}\setminus \{\emptyset \}}$ be the collection of all such simplices and $\x = (x_T)_{T\in 2^{[n]}\setminus \{\emptyset \}}$. Then the \emph{generalized permutohedron} is \[\x\P:=\sum_{T \in 2^{[n]} \setminus \{\emptyset\}} x_T \Delta_T \subseteq H_{\langle\x,\one\rangle}.\]
\end{defin}
%In this section, we use different ways to calculate the hyperplane dual mixed volume of the generalized permutohedron to obtain identities. Our main focus is on $\tilde m_{\P}(\x,\z)$ in the case when $\langle\x,\one\rangle=0$ and $\z=\mathbf{0}$. 
We give the following explicit formula (\Cref{prop:genperm}) for the (hyperplane) dual mixed volume of a generalized permutohedron.  

\begin{proposition}\label{prop:genperm}
When $\langle\x,\one\rangle=0$, or equivalently, substituting $x_{[n]}$ with $-\sum_{T\in 2^{[n]}\setminus \{[n]\}}x_T$, the hyperplane dual mixed volume of the generalized permutohedron becomes \[\tilde m_{\P}(\x)|_{\langle\x,\one\rangle=0}=(-1)^{n-1}\sum_{\sigma\in S_n}\prod_{a=1}^{n-1}\frac{1}{\displaystyle\sum_{\emptyset\neq T\subseteq\sigma[1:a]}x_T}.\]
\end{proposition} 

\begin{example}
Let $n = 3$.  Then we have
\begin{align*}
\tilde m_{\P}(\x)|_{\langle\x,\one\rangle=0} &= \frac{1}{x_1(x_1+x_2+x_{12})} +  \frac{1}{x_1(x_1+x_3+x_{13})} +  \frac{1}{x_2(x_1+x_2+x_{12})}  \\
&+  \frac{1}{x_2(x_2+x_3+x_{23})} +  \frac{1}{x_3(x_1+x_3+x_{13})} +  \frac{1}{x_3(x_2+x_3+x_{23})}.
\end{align*}
\end{example}

\TL{Is the multiple use of notation confusing?} \TL{Is the sign really there?}

\begin{thm}
When $k(\x) = 0$, $m_{P(\x)}$ is equal, up to sign, to the usual dual volume $\Vol^\vee(P(\x))$.
\end{thm}

\begin{proof}
We compute the hyperplane dual volume function $\Vol^\vee_\z(\Cone(P(\x))$, which is a rational function in $\x$ and $\z$.  Recall that $r_S \in \R^d$ is defined by
$$
\langle r_S, \y \rangle = \sum_{s \in S} y_s.
$$
Thus 
$$
\min_{\y \in \Delta_T}(\langle r_S, \y \rangle) = \begin{cases} 1 & \mbox{if $T \subseteq S$} \\
0 & \mbox{otherwise}.
\end{cases}
$$
We have
$$
h_{P(\x)}(r_S) = \sum_{T \subseteq S} x_T.
$$
The generators of $\Cone(P(\x))$ are given by
$$
r'_S = r_S - \alpha_{S}(\x) \one, \qquad \alpha_S(\x) := \frac{\sum_{T \subseteq S} x_T}{\sum_T x_T}.
$$
for $S \in 2^{[d]} \setminus \{\emptyset,[d]\}$.  By \eqref{eq:affinedual}, we have \TL{Check signs!}
$$
\Vol^\vee_\z(\Cone(P(\x))) = \sum_{\sigma \in S_d} \prod_{a=1}^{d-1} \frac{1}{\langle r'_{\sigma([a])},\z \rangle} = \sum_{\sigma \in S_d} \prod_{a=1}^{d-1} \frac{1}{z_{\sigma_1} + \cdots + z_{\sigma_a} - \alpha_{\sigma[a]}(\x)z_{[d]}},
$$
where $z_{[d]} := z_1 + z_2 + \cdots + z_d$.  Now, if $z_{[d]} = \sum_T x_T$, this simplifies to
$$
 \sum_{\sigma \in S_d} \prod_{a=1}^{d-1} \frac{1}{\langle r'_{\sigma([a])},\z \rangle} = \sum_{\sigma \in S_d} \prod_{a=1}^{d-1} \frac{1}{z_{\sigma_1} + \cdots + z_{\sigma_a} - \sum_{T \subseteq \sigma([1,a])} x_T}.
$$
Applying \Cref{prop:enrich}, this is equal to the dual volume of the polytope $P(\x) - \sum_i z_i e_i = P(\x')$ where 
$$
x'_T = \begin{cases} x_{\{i\}} - z_i & \mbox{if $T = \{i\}$,} \\
x_T & \mbox{otherwise.}
\end{cases}
$$
The condition that $z_{[d]} = \sum_T x_T$ turns into the condition that $P(\x') \subset H_0$, or equivalently, $k(\x') = 0$.
\end{proof}

\begin{cor}
Let $Q_1(\x),Q_2(\x),\ldots,Q_m(\x)$ be a mixed subdvision of $P(\x)$.  Then we have
$$
m_{P(\x)} = \sum_{i=1}^m m_{Q_i(\x)}.
$$
\end{cor}
\begin{proof}
Follows from \Cref{thm:affinevaluation}.
\end{proof}

One can apply the Cayley trick as in \Cref{sec:Cayley} to the generalized permutohedra and obtain fine mixed subdivisions of $\P$. We use results of Postnikov \cite{postnikov-permutohedra} for our computation of the dual mixed volume of the generalized permutohedra.

Let $N=2^n-1$ and let $\{T_1,\ldots,T_N\}$ be all nonempty subsets of $[n]$. Let $K_{N,n}$ be the complete bipartite graph with vertices in one part are labeled $i=1,2,\ldots,N$, and vertices the other part are $\bar j=\bar 1,\bar2,\ldots,\bar n$. Let $G\subseteq K_{N,n}$ be the subgraph where the vertex $i$ is connected to $\bar j$ if $j\in T_i$. For a collection of subsets $\J=(J_1,\ldots,J_N)$ where $J_i\subseteq T_i$, let $G_{\J}$ be the subgraph of $G$ where vertex $i$ is connected to $\bar j$ if $j\in J_i$.
\begin{lemma}\cite[Lemma 14.7]{postnikov-permutohedra}
For any fine mixed subdivisions of $\P$, its fine mixed cells are in bijection with nonempty subsets $\J=(J_1,\ldots,J_N)$ such that $G_{\J}$ is a spanning tree of $G$. In this case, the corresponding fine mixed cell of $\P$ is $\Q_{\J}:=(\Delta_{J_i})_{i=1}^N$.
\end{lemma}
When $G_{\J}$ is a spanning tree, removing the vertex $i\in[N]$ and the edges incident to it in the graph $G_{\J}$ results in $|J_i|$ connected components, each containing exactly one element in $J_i$. For an edge $(i,\bar j)$ in $G_{\J}$, denote this connected component that contains $\bar j$ as $A_{i,\bar j}\sqcup \bar A_{i,\bar j}$ where $A_{i,\bar j}\subseteq[N]$ and $\bar A_{i,\bar j}\subseteq[\bar n]=\{\bar1,\bar2\ldots,\bar n\}$. Define \[h^{\J}_{i,\bar j}(\x):=\sum_{k\in A_{i,\bar j}}x_{T_k}.\]
In other words, $x_{T_k}$ is in the summation for $h^{\J}_{i,j}(\x)$ if and only if the path between $k$ and $i$ in $G_{\J}$ passes through $\bar j$.
Note that 
\[\sum_{j\in J_i}h^{\J}_{i,\bar j}(\x)=\sum_{T_i\neq T\in 2^{[n]}\setminus\emptyset}x_T.\]

\begin{proposition}\label{prop:permutohedra-cell-dual-volume}
Let $\J$ be a collection of subsets so that $G_{\J}$ is a spanning tree of $G$. When $\langle\x,\one\rangle=0$, the hyperplane dual mixed volume of the fine mixed cell $\Q_{\J}$ is \[\tilde m_{\Q_{\J}}(\x)|_{\langle\x,\one\rangle=0}=(-1)^{n-1}\prod_{i:|J_i|>1}\frac{-x_{T_i}}{\prod_{j\in J_i}h_{i,\bar j}^{\J}(\x)}.\]
\end{proposition}

One very nice property of the formula given in \Cref{prop:permutohedra-cell-dual-volume} is that every factor $(-x_{T_i})/\prod_{j\in J_i}h_{i,\bar j}^{\J}(\x)$ equals a subtraction-free rational function once we have substituted $x_{[n]}=-\sum_{T\subset[n]}x_T$. This is because if $T_i=[n]$, then the numerator $-x_{T_i}=-x_{[n]}$ becomes positive and none of the $h_{i,\bar j}^{\J}(\x)$ contains $x_{[n]}$; and if $T_i\neq[n]$, then the numerator is negative but there is exactly one $h_{i,\bar j}^{\J}(\x)$ in its denominator with $x_{[n]}$ in it and thus the negative signs cancel. 

\end{comment}



% \subsection{Dual mixed Eulerian functions}
% Write permutohedron as Minkowski sum of hypersimplices.

\section{Associahedra}\label{sec:associahedra}


We give an explicit formula (\Cref{prop:assoc}) for the dual mixed volume of an \emph{associahedron} and relate it to the planar $\phi^3$-amplitude.

\begin{defin}\label{def:associahedra}
For $n\in\mathbb{Z}_{>0}$, let $\P=(\Delta_{[i,j]})_{1\leq i\leq j\leq n}$. The \emph{associahedron} is \[\x\P:=\sum_{1\leq i\leq j\leq n}x_{ij}\Delta_{[i,j]},\]
which is a special case of the generalized permutohedron.
\end{defin}
Note that here we use $x_{ij}$, where $1\leq i\leq j\leq n$ to index a variable. \Cref{def:associahedra} is commonly referred to as the \emph{Loday realization} \cite{loday} of the associahedra. Our presentation of the material largely follows \cite{postnikov-permutohedra}. 

%\begin{lemma}[Corollary 8.2 of \cite{postnikov-permutohedra}] Let $x_{ij}>0$ for $1\leq i\leq j\leq n$. The set of vertices of $\x\P$ is in bijection with $\PB(n)$, the set of all \emph{plane binary trees} with $n$ nodes. %Specifically, given $B\in \PB(n)$, denote its corresponding vertex as $\p_B$. Then its $k$-th coordinate equals \[\langle\p_B,\e_k\rangle=\sum_{l_B(k)\leq i\leq k\leq j\leq r_B(k)}x_{ij}.\]\end{lemma}

We need some additional notations in order to describe the normal fan $\N(P)$ of the associahedron $\x\P$. Let $B$ be a plane binary tree and let $e\in B$ be an edge. Deleting $e$ from this tree results in two connected components $[n]=L_{B,e}\sqcup U_{B,e}$ where $U_{B,e}$ contains the root of $B$. 
\begin{comment}
Let $\v_{B,e}$ be the indicator vector of $L_{B,e}$. In other words, the $k$-th coordinate of $\v_{B,e}$ equals $1$ if $k\in L_{B,e}$, and $0$ if $k\in U_{B,e}$. Let $\tilde\v_{B,e}$ be the projection of $\v_{B,e}$ onto $H=\{\y\in\R^n\:|\:\langle\y,\one\rangle=0\}$.

The following result can be deduced from \cite[Proposition 2.6]{loday}.

\begin{prop}
    \label{prop:normal-fan-associahedron}
The maximal cones in the normal fan of the associahedron $\x\P$ are in bijection with $\PB(n)$. Specifically, given $B\in \PB(n)$, its corresponding maximal cone can be written as $C_{B}:=\mathrm{span}_{\R_{\geq 0}}\{\tilde\v_{B,e}\:|\: e\in B\}$.
\end{prop}

\end{comment}

\begin{prop}\label{prop:assoc}
When $\langle\x,\one\rangle=0$, we have \[\tilde m_{\P}(\x)|_{\langle\x,\one\rangle=0}=(-1)^{n-1}\sum_{B\in\PB(n)}\prod_{e\in B}\frac{1}{\sum_{i,j\in L_{B,e}}x_{ij}}.\]
\end{prop}


%We now translate various statistics to planar cubic trees, by defining a bijection $\varphi:\PB(n)\rightarrow\PC(n+2)$ as follows. Each plane binary tree has a depth-first search ordering on its nodes, by traversing from its root and always going to the left child first (see \cite{postnikov-permutohedra} for details). Under this order, whenever we encounter a node with degree $d$, we turn it into an interior vertex and connect it to $3-d$ boundary vertices to form a planar cubic tree. If the root of $B$ has two children, then in $\varphi(B)$ it is connected to the boundary vertex labeled $n+2$; if the root of $B$ is $1$ which has only a right child, then in $\varphi(B)$ it is connected to the boundary vertices $n+1$ and $n+2$; if the root of $B$ is $n$ which has only a left child, then it $\varphi(B)$ it is connected to the boundary vertices $n+2$ and $1$; the rest of labels in $\varphi(B)$ is extended cyclically. An example is seen in \Cref{fig:plane-binary-tree-example}.

%For a planar cubic tree $T\in\PC(n)$ and an interior edge $e\in T$, removing $e$ from $T$ result in a partition of the boundary vertices by connectivity, denoted as $L_{T,e}\sqcup U_{T,e}$ where $n\in U_{T,e}$. By planarity, $L_{T,e}$ is an interval and $U_{T,e}$ is a cyclic interval. Under the map $\varphi$, edges of $B\in\PB(n)$ naturally correspond to interior edges of $\varphi(B)\in\PC(n+2)$.

\subsection{Relation to planar \texorpdfstring{$\phi^3$}{phi3}-amplitude at tree level}
We relate \Cref{prop:assoc} to a rational function $\mathcal{A}^{\phi_3}_n(\mathbf{s}_{ij})$ appearing in physics, called the $\phi^3$-amplitude; see \cite{AHLstringy,ABHY}.  This rational function was a main motivation for our study of dual mixed volumes.

Scattering amplitudes are functions that compute the outcome of scattering experiments in particle physics.  Traditionally, they are computed as the sum over Feynman diagrams which depend on the choice of particles and their interactions.  In ``planar $\phi^3$-theory'', the Feynman diagrams are \emph{planar cubic trees}.  For $n$-particle scattering, the amplitude\footnote{The full amplitude has a perturbative expansion; we only consider the first term which is a sum over trees.  The later terms involve graphs with cycles.} $\mathcal{A}^{\phi_3}_n(\mathbf{s}_{ij})$ is a sum over all planar cubic trees with $n$ nodes.

Comparing it with \Cref{prop:assoc}, we obtain the following.

\begin{prop}
Up to a sign, the dual mixed volume $\tilde m_{\P}(\x)|_{\langle\x,\one\rangle=0}$ of the associahedron is equal to the planar $\phi^3$-amplitude $\mathcal{A}^{\phi_3}_n$ under the substitution
$x_{ij} \mapsto s_{i,j+1}$.
\end{prop}






\acknowledgements{
We thank Alexander Barvinok for a number of enlightening conversations, especially in relation to \Cref{thm:valuation}.  T. Lam was supported by the Simons Foundation and by the National Science Foundation under grants DMS-1953852 and DMS-2348799.
}


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