\begin{document}

	\title{Centro-Affine Tensor Valuations}

	\begin{abstract}
		\noindent
		We completely classify all measurable $\SL$-covariant symmetric tensor valuations on convex
polytopes containing the origin in their interiors.
It is shown that essentially the only examples of such valuations are the moment tensor
and a tensor derived from $L_p$ surface area measures.
This generalizes and unifies earlier results for the scalar, vector and matrix valued case.
 \\[0.5cm]
		Mathematics subject classification: 52B45, 52A20
	\end{abstract}

	\section{Introduction}
		A map $\mu \colon \mathcal{S} \to \langle A, + \rangle$ defined on a collection of sets 
$\mathcal{S}$ with values in an abelian semigroup $\langle A, + \rangle$ is
called a valuation if
\[
	\mu ( P \cup Q ) + \mu ( P \cap Q ) = \mu ( P ) + \mu ( Q )
\]
whenever $P$, $Q$, $P \cup Q$, $P \cap Q \in \mathcal{S}$.

One of the most influential results from the classical Brunn-Minkowski theory is
\mbox{Hadwiger's} classification of continuous rigid motion invariant 
valuations $\mu \colon \cK^n \to \R$. Here, $\cK^n$ denotes the space of convex bodies,
i.e.\ non-empty compact convex subsets of $\R^n$ equipped with the Hausdorff metric.
Hadwiger showed that each such valuation is a linear combination of the intrinsic volumes.
The latter are of basic geometric nature and include volume, surface area, mean width and the Euler Characteristic.

Two fundamental quantities that are not covered by Hadwiger's theorem
are Blaschke's equi-affine and centro-affine surface area.
The latter is not translation invariant and in fact, does not even fit in the framework of the Brunn-Minkowski theory.
It does, however, belong to the so called $L_p$-Brunn-Minkowski theory,
which was shaped by Lutwak \cites{Lut93b, Lut96} in the mid 1990s.
It is based on Firey's $L_p$ addition of convex bodies
containing the origin in their interiors.
The set of all such convex bodies is denoted by $\cK_o^n$.
Since then, this theory has become a central part of modern convex geometry (see \cite{Sch13}*{Chapter 9}). 
The impact of the $L_p$ theory ultimately led to the discovery of an even more general framework:
The Orlicz-Brunn-Minkowski theory (see, e.g., \cites{HabLutYanZha09, garhugwei13, LutYanZha09jdg, 
LutYanZha09badv, ZhuZhouXu14, ZouXiong14, XiJinLeng14, GarHugWeiYe15, LR10, habpar13, Lud09}). 

A characterization of Blaschke's equi-affine and centro-affine surface area
was finally established in a landmark result by Ludwig and Reitzner in \cite{LR10},
where they classified the natural family of Orlicz affine surface areas.
However, one crucial part of the problem remained open
since one of their assumptions was a certain behavior of the maps on convex polytopes.
The first step to bridge this last gap had already been taken by Ludwig \cite{Lud02advval},
but the complete result was only established very recently by the authors \cite{habpar13}:

\begin{theorem}\label{AffHadwiger}
	A map $\mu \colon \cK_o^n\to\R$ is an upper semicontinuous $\SL$-invariant valuation if and only if 
	there exist constants $c_0,c_1, c_2\in\R$ and a function $\varphi\in\textnormal{Conc}(\R_+)$ 
	such that
	\[
		\mu (K)=c_0 \chi(K)+c_1 V(K)+c_2 V(K^*)+\Omega_{\varphi} (K)
	\]
	for all $K\in\cK_o^n$.
\end{theorem}
Here, $\chi$ denotes the Euler Characteristic, $V$ stands for volume, $K^*$ is the polar body
of $K$, and the $\Omega_{\varphi}$ are Orlicz affine surface areas. 
The reader is referred to Section \ref{prelim} and \cite{habpar13} for details. 

This centro-affine Hadwiger theorem has a discrete version for valuations defined on $\cP_o^n$, 
i.e.\ convex polytopes containing the origin in their interiors (see \cites{habpar13, HabParMoments}): 
A map $\mu \colon \cP_o^n\to\R$ is a measurable $\SL$-invariant valuation if and only if there exist constants 
$c_0, c_1, c_2 \in \R$ such that
	\[
		\mu (P)=c_0 \chi(P) + c_1 V(P) + c_2 V(P^*)
	\]
for all $P\in\cP_o^n$.

The aim of the present paper is to generalize this result to tensor valued valuations of arbitrary rank.
Such a generalization to the vector valued case was already established by Ludwig in \cite{Ludwig:moment},
where she characterized the moment vector, i.e.\ the centroid without volume normalization.
In a highly influential article, Ludwig \cite{Lud03} was also able to show the corresponding result for matrix valued valuations.
In both papers, she assumed compatibility with the whole \emph{general} linear group.
A version for the vector valued case that only assumes compatibility with the \emph{special} linear group
was very recently proved by the authors \cite{HabParMoments}.
The present article is the first one to establish a classification for tensor valuations of arbitrary rank in the context
of “centro-affine geometry”.

The study of tensor valuations became the focus of increased attention after Alesker's breakthrough \cite{Alesker99}.
The new techniques developed in this paper enabled him to prove a long sought after characterization
of the rigid motion compatible Minkowski tensors in \cite{Alesker00a}.
In recent years, tensor valuations were studied intensively 
(see, e.g., \cites{AleBerSchu, BerHug15, hugschsch08, Lud03, Lud11, HugSchLocTen, wannerer13, HabParMoments}). 
This is in part due to their applications in morphology and anisotropy analysis of cellular, granular or porous structures
(see, e.g., \cites{BeisArt02, SchrAl2011, SchAl2010, SchrAl2013}).

Let us give two examples of tensor valuations.
Write $\Sym{\R^n} \subseteq \Ten{\R^n}$ for the space of symmetric tensors of rank $p \in \{ 0, 1, \ldots \}$.
The first example is the moment tensor map $M^{p, 0} \colon \cP_o^n \to \Sym{\R^n}$. 
It is defined as
\[
	M^{p, 0}(P) = (n+p) \int_P x^{\odot p}\,dx ,
\]
where integration is with respect to Lebesgue measure and $\odot$ stands for the symmetric tensor product.
Ignoring constant factors, $M^{1, 0}(P)$ is the moment vector
and $M^{2, 0}(P)$ corresponds to the Legendre ellipsoid from classical mechanics.

The second example, $M^{0, p} \colon \cP_o^n \to \Sym{\R^n}$, is given by
\begin{equation}\label{eq: def M^{0,p}}
	M^{0, p}(P) = \int_{S^{n-1}} u^{\odot p}\, dS_p(P,u) .
\end{equation}
Here, $S^{n-1} \subseteq \R^n$ denotes the Euclidean unit sphere and $S_p(P, \cdot)$ is the
$L_p$ surface area measure of $P$ (see Section \ref{prelim} for details).
This tensor vanishes for $p = 1$
and corresponds to the Lutwak-Yang-Zhang ellipsoid from \cite{LutYanZha00duke} for $p = 2$.

We will prove that these are essentially the only examples of tensor valuations 
which are compatible with the $\SL$.
This compatibility is contained in the following definition.
The group $\GL$ acts naturally on $\Ten{\R^n}$	by
\[
	\phi \cdot x = \phi^{\otimes p} (x) 
\]
for all $\phi \in \GL$ and $x \in \Ten{\R^n}$.
A map $\mu \colon \cP_o^n \to \Sym{\R^n}$ is called $\SL$-covariant if
	\[	
		\mu ( \phi P ) = \phi \cdot \mu ( P )
	\]
for all $\phi \in \SL$ and each $P \in \cP_o^n$.	
Moreover, $\mu$ is called measurable if it is Borel measurable.
We are now in a position to state our main result in dimensions greater or equal than three.
	
\begin{theorem}\label{th: MainN}
	Let $p \geq 2$ and $n \geq 3$. 
	A map $\mu \colon \cP_o^n \to \Sym{\R^n}$ is a measurable
	$\SL$-covariant valuation if and only if there exist constants $c_1, c_2 \in \R$ such that
	\[
		\mu(P) = c_1 M^{p, 0}(P) + c_2 M^{0, p}(P^*)
	\]
	for all $P \in \cP_o^n$.
\end{theorem}

The $L_p$ surface area measure appearing in \eqref{eq: def M^{0,p}} is a central notion of the $L_p$-Brunn-Minkowski theory. 
One of the major problems in the field, the so called $L_p$ Minkowski problem, asks which measures are $L_p$ surface area measures 
(see, e.g., \cites{BorLutYanZha, ChoWan06, Lut93b, Zhu2015}). 
Moreover, $L_p$ surface area measures found applications in such diverse fields as affine isoperimetric inequalities
(see, e.g., \cites{CamGro02adv, HS09jdg, LutYanZha00jdg}), Sobolev inequalities (see, e.g., \cites{HSX, CLYZ, LudXiaZha09}),
valuation theory (see, e.g., \cites{Schduke, Hab11, Lud05, Par10a, SchWa15}), 
and information theory (see, e.g., \cites{LutYanZha02duke, Lud11, Lutal2012}).

In \cite{Lut93b}, Lutwak introduced $L_p$ surface area measures in connection with Firey's $L_p$ addition of convex bodies.
In some way or the other, the occurrence of these measures is usually related to this $L_p$ addition.  
We want to emphasize that by Theorem \ref{th: MainN}, $L_p$ surface area measures naturally appear in a completely different context.
This clearly underlines the basic character of these measures. 
For an axiomatic characterization of $L_p$ surface area measures themselves, we refer to \cite{habpar13a}.

The operators occurring in Theorem \ref{th: MainN} are special members of the following family of valuations. 
For $r, s \in \{ 0, 1, \ldots \}$ with $r + s = p$ set
\[
	M^{r, s}(P) = \int_{\partial P} x^{\odot r} \odot u_x^{\odot s} \, h_P^{1-s}(u_x) \, d\mathcal{H}^{n-1}(x) . 
\]
Here, $h_P$ denotes the support function of $P$ and 
integration is with respect to $(n-1)$-dimensional Hausdorff measure over the boundary $\partial P$ of $P$.
The vector $u_x$ is the outer unit normal vector of $P$ at the boundary point $x$. 
Note that $\mathcal{H}^{n-1}$ almost all boundary points have a unique outer unit normal $u_x$.
To the best of our knowledge, these operators have not yet been studied in general.

Except for the extreme cases $M^{p, 0}$ and $M^{0, p} \circ \, ^*$, the members of the above family are not $\SL$-covariant.
However, in the plane they can be modified in a simple way so that they possess this $\SL$-covariance.
Indeed, we denote by $\rho$ the counter-clockwise rotation about an angle of $\frac \pi 2$ and set
\[
	M^{r, s}_{\rho}(P) = \int_{\partial P} x^{\odot r} \odot (\rho u_x)^{\odot s} \, h_P^{1-s}(u_x) \, d\mathcal{H}^1(x) . 
\]
The planar version of Theorem \ref{th: MainN} then reads as follows.

\begin{theorem}\label{th: Main2}
	Let $p \geq 2$.
	A map $\mu \colon \cP_o^2 \to \Sym{\R^2}$ is a measurable
	$\SL[2]$-covariant valuation if and only if there exist constants $c_0, \ldots, c_{p-2}, c_p, c_{p+1} \in \R$ such that
	\[
		\mu(P) = \sum_{\substack{i=0 \\ i \neq p-1}}^p c_i M_{\rho}^{i, p-i}(P) + c_{p+1} \rho \cdot M^{p, 0}(P^*)
	\]
	for all $P \in \cP_o^2$.
\end{theorem}

Denote by $\ValCo{p}{n}$ the vector space of measurable $\SL$-covariant valuations $\mu \colon \cP^n_o \to \Sym{\R^n}$.
As explained above, a complete classification of $\ValCo{0}{n}$ was established in \cites{habpar13, HabParMoments},
whereas the description of $\ValCo{1}{n}$ can be found in \cite{HabParMoments}.
In order to get a complete picture, we finally summarize these results and the main theorems of 
the present article.
\begin{theorem}\label{th: n-dim, SL(n)-covariant}
	For $n \geq 3$ the following holds.
	\begin{itemize}
		\item
		A basis of $\ValCo{0}{n}$ is given by $\chi$, $V$ and $V \circ \, ^*$.

		\item
    A basis of $\ValCo{1}{n}$ is given by $M^{1, 0}$.

		\item
    For $p \geq 2$, a basis of $\ValCo{p}{n}$ is given by $M^{p, 0}$ and $M^{0, p} \circ \, ^*$.
	\end{itemize}
\end{theorem}
As was mentioned before, the planar case is different and needs to be treated seperately.
\begin{theorem}\label{th: 2-dim, SL(2)-covariant}
	For $n = 2$ the following holds.
	\begin{itemize}
		\item
		A basis of $\ValCo{0}{2}$ is given by $\chi$, $V$ and $V \circ \, ^*$.

		\item
    For $p \geq 1$, a basis of $\ValCo{p}{2}$ is given by $M^{i, p-i}_\rho$ for
    $i \in \{ 0, \ldots, p \} \setminus \{ p-1 \}$ and $\rho \cdot M^{p, 0} \circ \, ^*$.
	\end{itemize}
\end{theorem}
In this paper, we will actually provide a unified proof of Theorems \ref{th: n-dim, SL(n)-covariant}
and \ref{th: 2-dim, SL(2)-covariant}.
Therefore, we also provide new proofs of the results in \cite{habpar13} and \cite{HabParMoments} which fit into the general
context of tensor valuations with arbitrary rank.

	\section{Notation and preliminary results}\label{prelim}
		For later use, we collect in this section notation and basic facts.
Well known results about convex bodies are stated without references.
We refer the reader to the excellent books of 
Gardner \cite{Gar95}, Gruber \cite{Gruber:CDG}, and Schneider \cite{Sch13}
for more information.

Let us begin with two one-dimensional facts.
The first one is the solution to Cauchy's functional equation. 
As is well known, the only measurable functions $f \colon \R \to \R$ which satisfy
\[
	f(x + y) = f(x) + f(y)
\]
for all $x, y \in \R$ are the linear ones.
The same holds for functions $f \colon (0, \infty) \to \R$ and $f \colon \R^n \to \R$.
The second one is a version of Vandermonde's identity,
\begin{equation}\label{eq: Vandermonde}
	\sum_{j=0}^i \binom {-\frac p 2} {i-j} \binom {\frac p 2} {j} = 0
\end{equation}
for $i \geq 1$.
This follows from the equality $(1 + x)^{-\frac p 2}(1 + x)^{\frac p 2} = 1$
by comparing coefficients of the Taylor expansions of the involved functions.

Now, we turn towards higher dimensions.
The space $\R^n$, $n \geq 1$, will be equipped with the standard inner product and the norm induced by it.
Denote by $e_1,\ldots, e_n \in \R^n$ the canonical basis vectors and
write $S^{n-1}$ for the set of all unit vectors with respect to this norm.

Throughout this paper, we fix the standard basis of $\Ten{\R^n}$ induced by the canonical basis vectors $e_1, \ldots, e_n$.
For tensors $x_1, \ldots, x_p \in \Ten{\R^n}$, their symmetric tensor product is defined as 
\[
	x_1 \odot \cdots \odot x_p = \frac {1}{p!} 
	\sum_{\sigma \in \mathfrak{S_p}} x_{\sigma(1)} \otimes \cdots \otimes x_{\sigma(p)} ,
\]
where $\mathfrak{S_p}$ denotes the symmetric group of $\{1, 2, \ldots, p\}$.
Note that the normalization is chosen in such a way that $x \odot \cdots \odot x = x \otimes \cdots \otimes x$.
Let $K \in \Ten{\R^2}$ and $\alpha \in \{1, 2\}^p$ be a multiindex.
If $\phi \in \GL[2]$, then the action of $\phi$ on $K$ can be written as
\begin{equation}\label{eq: action on tensors}
	\phi \cdot K = \sum_{\alpha} \sum_{\beta} K_\beta \phi_{\alpha_1 \beta_1} \cdots \phi_{\alpha_p \beta_p} 
	               e_{\alpha_1} \otimes \cdots \otimes e_{\alpha_p} .
\end{equation}
Here, $K_\beta$ denote the coefficients of $K$ with respect to the basis we fixed before.
Multiindices will be viewed as being equipped with their standard partial order.
Therefore, we immediately arrive at	
	\begin{equation}\label{eq: tilt expanded}
  	\left[ \tilt z K \right]_{\alpha} = \sum_{\beta \geq \alpha} 
  	K_{\beta} z^{\vert\{i \colon \beta_i = 2\}\vert - \vert\{i \colon \alpha_i = 2\}\vert}.
  \end{equation}
For the space $\Sym{\R^2}$ of symmetric tensors we fix the basis 
\[
	e_1^{\odot p-i} \odot e_2^{\odot i} , \qquad i = 0, \ldots, p .
\]
Let $K \in \Sym{\R^2}$.
The coordinates of $K$ with respect to this basis are denoted by $K_i$.
For even $p$ and a $\phi \in \GL[2]$ we therefore have
\begin{equation}\label{eq: pth component middle term}
	\left[ \phi \cdot e_1^{\odot \frac p 2} \odot e_2^{\odot \frac p 2} \right]_p 
	= ( \phi_{2 \! 1} \phi_{2 2} )^{\frac p 2} .
\end{equation}
Relation \eqref{eq: tilt expanded} and a straightforward computation prove
\begin{equation}\label{eq: tilt expanded symmetric}
 	\left[ \tilt z K \right]_i = \sum_{j=i}^p \binom j i K_j z^{j-i} .
\end{equation}
This in turn yields
  \begin{equation}\label{eq: tilt expanded symmetric transpose}
  	\left[ \ptwotwo 1 z 0 1 \cdot K \right]_i = \sum_{j=0}^i \binom {p-j} {p-i} K_j z^{i-j} .
  \end{equation}
Let us briefly discuss a tensor version of Cauchy's functional equation.
Let $F \colon \R^n \to \Ten{\R^n}$ be a measurable function with
\[
	F(x + y) = F(x) + F(y)
\]
for all $x, y \in \R^n$.
Using the scalar Cauchy equation, it is not hard to show that the component functions of $F$ are linear.
Interpreting tensors as multilinear maps, it therefore follows that there exists an $\tilde F \in \Ten[p+1]{\R^n}$ 
such that
\begin{equation}\label{eq: tensor cauchy}
	F(x) (v_1, \ldots, v_p) = \tilde F (v_1, \ldots, v_p, x)
\end{equation}
for all $v_1, \ldots, v_p, x \in \R^n$.
In other words, $F$ can be interpreted as an element of $\Ten[p+1]{\R^n}$.

Next, we collect some facts about tensor integrals.
As usual, integrals over tensors are defined componentwise.
Thus, a straightforward calculation in combination with \eqref{eq: action on tensors}
proves for a continuous function $F \colon [a,b] \to \Ten{\R^n}$ that 
	\begin{equation}\label{eq: integral compatible SL(n)}
		\int_a^b \phi \cdot F(x) \,dx = \phi \cdot \int_a^b F(x) \,dx 
	\end{equation}
for all $\phi \in \GL$.
If $F \colon \R \to \Ten{\R^n}$ is continuous, then one can check the symmetry of its images 
by looking at certain integrals. Indeed, by the componentwise definition of the integral and 
differentiation we have
	\begin{equation}\label{eq: int symmetric}
		F(x) \in \Sym{\R^n} \quad \textnormal{for all } x \in \R
		\Longleftrightarrow 
		\int_0^x F(z) \,dz \in \Sym{\R^n} \quad \textnormal{for all } x \in \R .
	\end{equation}
We conclude our treatment of integrals with the following injectivity type properties.
For $K \in \Ten{\R^2}$, we infer from \eqref{eq: tilt expanded} that 
	\begin{equation}\label{eq: integral tilt kernel}
	  \int_0^x \tilt z K \,dz = 0 \quad \textnormal{for some } x \in \R \setminus \{ 0 \}
	  \quad \Longleftrightarrow \quad K = 0 .
	\end{equation}
Hence,
\[
	K \mapsto \int_0^1 \tilt z K \ dz
\]
is a linear isomorphism on $\Ten{\R^2}$.
A variant of this implication is
	\begin{equation}\label{eq: integral tilt kernel transpose}
	  \int_0^x \ptwotwo 1 {-z} 0 1 K \,dz = 0 \quad \textnormal{for some } x \in \R \setminus \{ 0 \}
	  \quad \Longleftrightarrow \quad K = 0 .
	\end{equation}

Let a convex polytope $P \in \cP_o^n$ be given.
In the next paragraph we recall some basic geometric quantities associated with $P$.
The first example is the support function $h_P$.
This is the function $h_P \colon \R^n \to \R$ defined by
\[
	h_P(x) = \max \{ x \cdot y: y \in P \}.
\]
The polar body $P^*$ of $P$ is given by
\[
	P^* = \{ x \in \R^n: x \cdot y \leq 1 \text{ for all } y \in P \} .
\]
Note that for each $\phi \in \GL$ we have
\begin{equation}\label{eq: polarity contravariant}
	(\phi P)^* = \phi^{-t} P^*,
\end{equation}
where $\phi^{-t}$ denotes the inverse of the transpose of $\phi$.
In particular,
\begin{equation}\label{eq: polar homogeneity}
	(\lambda P)^* = \lambda^{-1} P^*
\end{equation}
for all positive $\lambda$.
We define the polarity map $ ^* \colon \cP_o^n \to \cP_o^n$ 
as the function which assigns to each polytope its polar body.
It is well known that this map is a homeomorph involution.
The surface area measure $S(P, \cdot)$ is defined for each Borel set $\omega \subseteq S^{n-1}$ as
\[
	S(P,\omega) = \mathcal{H}^{n-1}\{ x \in \partial P: 
	\exists \text{ an outer unit normal } u_x \text{ at } x \text{ which belongs to } \omega \}.
\] 
Surface area measures have their centroid at the origin, i.e.
\begin{equation}\label{eq: surface centroid}
	\int_{S^{n-1}} u \ dS(P, u) = 0
\end{equation}
for all $P \in \cP_o^n$.
The $L_p$ surface area measure $S_p(P, \cdot)$ is given by
\[
	S_p(P, \omega) = \int_{\omega} h_P^{1-p}(u) \ dS(P, u). 
\]

Next, we will generalize the concept of $\SL$-covariance a little bit.
We write $\VL$ for the set of linear maps having determinant either $1$ or $-1$.
Let $\varepsilon \in \{0,1\}$ and $G \subseteq \VL$ be given.
A map $\mu \colon \cP^n_o \to \Sym{\R^n}$ is said to be $G$-$\varepsilon$-covariant 
or $\varepsilon$-covariant with respect to $G$ if
	\[
		\mu (\phi P) = (\det \phi)^{\varepsilon} \phi \cdot \mu (P)
	\]
for every $P \in \cP^n_o$ and each $\phi \in G$.
In order to simplify the notation in the sequel, we write $\ValCoEps{p}{n}$
for the vector space of measurable $\VL$-$\varepsilon$-covariant valuations.

Let $\mu \in \ValCo{p}{n}$.
Choose a $\theta \in \VL \setminus \SL$.
For all $P \in \cP^n_o$ define
\[
	\mu^0 (P) = \frac 1 2 \left( \mu (P) + \theta \cdot \mu (\theta^{-1} P) \right)
\]
and
\[
	\mu^1 (P) = \frac 1 2 \left( \mu (P) - \theta \cdot \mu (\theta^{-1} P) \right).
\]
The $\SL$-covariance of $\mu$ implies that these definitions do not depend on the choice of $\theta$.
Clearly, $\mu^0$ and $\mu^1$ are measurable valuations and $\mu = \mu^0 + \mu^1$.
Moreover, it is easy to see that $\mu^0 \in \ValCoZero{p}{n}$ and $\mu^1 \in \ValCoOne{p}{n}$.
Hence,
\begin{equation}\label{eq: Val direct sum}
	\ValCo{p}{n} = \ValCoZero{p}{n} \oplus \ValCoOne{p}{n}.
\end{equation}

The convex hull of a set $A \subseteq \R^n$ is written as $[A]$.
In the context of double pyramids, the following symbols will usually have a fixed meaning.
The letters $a, b, c, d$ will denote positive real numbers
with associated line segments $I := [-a e_1, b e_1]$ and $J := [-c e_n, d e_n]$, respectively.
The letters $x, y$ will denote elements of $\R^{n-1}$.
In particular, for $n = 2$ we have $J = [-c e_2, d e_2]$ and $x,y \in \R$.
The letter $B$ will denote an element of $\cP^{n-1}_o$.
For $n = 2$, we say that $a, b, c, d, x, y$ form a double pyramid if
\[
	\left[ I, -c \ptwo x 1, d \ptwo y 1 \right] \cap e_2^\perp = I
\]
and for $n \geq 3$, we say that $B, c, d, x, y$ form a double pyramid if
\[
	\left[ B, -c \ptwo x 1, d \ptwo y 1 \right] \cap e_n^\perp = B .
\]
If $x = y = 0$, then we call the double pyramid straight.
The set of double pyramids will be denoted by $\cR^n$
and the set of straight double pyramids by $\cQ^n$.
In \cite{Lud02advval}, Ludwig proved that if a real valued valuation $\mu \colon \cP^n_o \to \R$ 
vanishes on all $\SL[n]$-images of elements in $\cR^n$, then it vanishes on $\cP_o^n$.
A componentwise application of this fact yields the following result.

\begin{theorem}\label{th: determined by values on SL(n)(R^n)}
	Let $n \geq 2$.
	Suppose that $\mu \colon \cP^n_o \to \Ten{\R^n}$ is a valuation which vanishes on all $\SL[n]$-images of elements in $\cR^n$.
	Then $\mu$ vanishes everywhere.
\end{theorem}
Clearly, for $n = 2$, straight double pyramids can be split into two triangles having one side contained
in $e_2^{\bot}$. The set of such straight triangles 
	\[
		[I, -c e_2] \quad \text{ and } \quad [I, d e_2]	
	\]
is denoted by $\cT^2$.

We need an explicit description of $M_\rho^{i, p-i}$ on straight double pyramids.
We start with the following calculation.
Note that in the next two lemmas we use the common convention that
\begin{equation}\label{eq: convention for binomial coefficient}
	\binom i j = 0 \quad \text{if $j < 0$ or $j > i$} .
\end{equation}

\begin{lemma}\label{le: M^{j,p-j} help}
	For $b, c > 0$ and $i \in \{ 0, \ldots, p\}$, we have
	\begin{equation} \label{eq: int b c}
		(i+1) (bc)^{1-p+i} \int_0^1 {\ptwo {bt} {-c(1-t)}}^{\odot i} \odot {\ptwo b c}^{\odot p-i} \ dt = \\
		\sum_{l = 0}^p m_{i,l} b^{1+i-l} c^{1-p+i+l} e_1^{\odot p-l} \odot e_2^{\odot l} ,
	\end{equation}
	where
	\[
		m_{i,l} = \binom {p-i-1} l + (-1)^i \binom {p-i-1} {l-i-1}
	\]
	for $i \neq p$ and
	\[
		m_{p,l} = (-1)^l .
	\]
	In particular, for $i \neq p$, $m_{i,l} = 0$ if $p-i \leq l \leq i$.
\end{lemma}

\begin{proof}
	Define $L$ as the left hand side of \eqref{eq: int b c}.
	Writing the first tensor product in coordinates,
	using well known results for the Beta function,
	and writing the second tensor product in coordinates,
	we calculate
	\begin{align*}
		L
		&= (i+1) \sum_{j = 0}^i \binom i j (-1)^j b^{1-p+2i-j} c^{1-p+i+j} \int_0^1 t^{i-j} (1-t)^j \ dt \
		e_1^{\odot i-j} \odot e_2^{\odot j} \odot {\ptwo b c}^{\odot p-i} \\
		&= \sum_{j = 0}^i (-1)^j b^{1-p+2i-j} c^{1-p+i+j}
		e_1^{\odot i-j} \odot e_2^{\odot j} \odot {\ptwo b c}^{\odot p-i} \\
		&= \sum_{j = 0}^i \sum_{k = 0}^{p-i} \binom {p-i} k (-1)^j b^{1+i-j-k} c^{1-p+i+j+k}
		e_1^{\odot p-j-k} \odot e_2^{\odot j+k} .
	\end{align*}
	Summing first over $l = j+k$ and keeping convention \eqref{eq: convention for binomial coefficient}
	for the binomal coefficient in mind, this becomes
	\[
		L = \sum_{l = 0}^p \sum_{j = 0}^i \binom {p-i} {l-j} (-1)^j b^{1+i-l} c^{1-p+i+l}
		e_1^{\odot p-l} \odot e_2^{\odot l} .
	\]
	For $i = p$, we clearly have
	\[
		\sum_{j = 0}^p \binom 0 {l-j} (-1)^j = (-1)^l .
	\]
	Assume $i \neq p$.
	A well known formula for an alternating sum of binomal coefficients states
	\[
		\sum_{j = 0}^l \binom {p-i} j (-1)^j = (-1)^l \binom {p-i-1} l .
	\]
	Note that $p-i \geq 1$ and that we again use convention \eqref{eq: convention for binomial coefficient}
	for the binomal coefficient from above.
	Hence,
	\begin{align*}
		\sum_{j = 0}^i \binom {p-i} {l-j} (-1)^j
		&= (-1)^l \sum_{j = l-i}^l \binom {p-i} j (-1)^j \\
		&= \binom {p-i-1} l + (-1)^i \binom {p-i-1} {l-i-1} .
	\end{align*}
\end{proof}

With the aid of this result, we can now calculate $M_\rho^{i, p-i}$ on
straight double pyramids.

\begin{lemma}\label{le: M on crosspolytopes}
	For $i \in \{ 0, \ldots, p\}$, we have
	\begin{multline*}
		M_\rho^{i, p-i}[I,J]
		= \frac 1 {i+1} \sum_{l = 0}^p m_{i,l} \left[
		(-1)^{i+l} a^{1+i-l} c^{1-p+i+l}
		+ b^{1+i-l} c^{1-p+i+l}
		\right. \\ \left.
		+ (-1)^p a^{1+i-l} d^{1-p+i+l}
		+ (-1)^{p+i+l} b^{1+i-l} d^{1-p+i+l}
		\right] e_1^{\odot p-l} \odot e_2^{\odot l}
	\end{multline*}
	for all $a, b, c, d > 0$, where $m_{i,l}$ is defined as in Lemma \ref{le: M^{j,p-j} help}.
	In particular, for $p \geq 1$, $M_\rho^{p-1, 1}[I,J] = 0$.
\end{lemma}

\begin{proof}
	Clearly, the double pyramid $[I,J]$ has four edges.
	Hence, the defining integral of $M_\rho^{i, p-i}$ can be split into four integrals along these line segments.
	Let us consider the edge $[b e_1, -c e_2]$.
	Using the parametrization 
	\[
		\gamma(t) = t b e_1 - (1-t)c e_2, \quad t \in [0, 1],
	\]
	an elementary calculation shows that 
	\[
		\int_{[b e_1, -c e_2]} x^{\odot i} \odot (\rho u_x)^{\odot p-i} \ d\mathcal H^1(x)
	\]
	equals, up to a factor of $i+1$, the integral considered in Lemma \ref{le: M^{j,p-j} help}.
	Similar observations are true for the other three edges.
	Summing the expressions from Lemma \ref{le: M^{j,p-j} help} for all edges yields the desired result.
	Finally, for $i \neq p$, note that the terms for $l = i+1$ as well as $l = p-i-1$ cancel out.
\end{proof}

Let $B = [I,J]$. Note that $B^* = [- a^{-1}, b^{-1}] \times [- c^{-1}, d^{-1}]$. 
Thus,
\[
	\left[ M^{p,0}(B^*) \right]_p = 
	\frac{1}{p+1} \left( a^{-1} + b^{-1} \right)\left( d^{-p-1} + (-1)^p c^{-p-1} \right).
\]
In combination with the last lemma, we see that
\begin{equation}\label{eq: M(B) neq 0}
	\left[ M_\rho^{i,p-i} (B) \right]_0 , \quad
	i \in \{0,\ldots,p\}\backslash\{p-1\} , \quad \text{and} \quad
	\left[\rho \cdot M^{p,0}(B^*)\right]_0
\end{equation}
do not vanish for all double pyramids $B$.
It follows immediately from their definition, that the $M_\rho^{i,p-i}$ are
homogeneous, i.e.
\begin{equation}\label{eq: M homogeneous 2 dim}
	M_\rho^{i,p-i}( \lambda P )= \lambda^{2-p+2i} M_\rho^{i,p-i}(P)
\end{equation}
for all $P \in \cP_o^2$ and $\lambda > 0$.
Combining the last two facts and \eqref{eq: polar homogeneity}, it is easy to see that the family 
\[
	M_\rho^{i,p-i}, i \in \{0,\ldots,p\}\backslash\{p-1\}, \quad \rho \cdot M^{p,0} \circ \, ^* 
\]
is linearly independent.

We also state a few simple facts about $M^{i,p-i}$ in dimension $n \geq 2$ for later reference.
Clearly, these maps are homogeneous,
\begin{equation}\label{eq: M homogeneous n dim}
	M^{i,p-i}( \lambda P )= \lambda^{n-p+2i} M^{i,p-i}(P)
\end{equation}
for all $P \in \cP_o^n$ and $\lambda > 0$.
The $e_1^{\odot p}$-coordinates of
\begin{equation}\label{eq: M(B) neq 0, n dim}
	M^{p, 0}(B^*) \quad \text{and} \quad M^{0, p}(B)
\end{equation}
do not vanish for all crosspolytopes $B$ except for $M^{0,1}(B)$.
To see this, one can look at crosspolytopes that are sufficiently asymmetric with respect to $e_1^\perp$.
Combining the last two facts and \eqref{eq: polar homogeneity}, it is easy to see that
\[
	M^{p, 0} \circ \, ^* \quad \text{and} \quad M^{0, p}
\]
are linearly independent.

Next, we show that $M_\rho^{i,p-i}$, for $n = 2$, and $M^{i,p-i}$, for $n \geq 2$, are valuations.
The easiest way to see this is to write them as integrals over the support measure $\Lambda_{n-1}$ (see \cite{Sch13}*{Chapter 4}).
For the latter, we have
\[
	M^{i,p-i}(P) = 2 \int_{\R \times S^{n-1}} x^{\odot i} \odot u^{\odot p-i} \, (x \cdot u)^{1-p+i} \, d\Lambda_{n-1}(P, (x,u))
\]
for all $P \in \cP^n_o$.
Now, the valuation property, the covariance properties, as well as the continuity of these maps
follow from similar properties of the support measure $\Lambda_{n-1}$.
In particular,
\begin{equation}\label{eq: M_rho signum covariant}
	M_{\rho}^{i,p-i}( \phi P )= (\det \phi)^{p-i} \ \phi \cdot M_{\rho}^{i,p-i}(P) ,
\end{equation}
for all $P \in \cP_o^2$ and $\phi \in \VL[2]$.
Furthermore,
\begin{equation}\label{eq: M covariant, n dim}
	M^{p,0}( \phi P )= \phi \cdot M^{p,0}(P) \quad \text{and} \quad
	M^{0,p}( (\phi P)^* ) = \phi \cdot M^{0,p}( P^* )
\end{equation}
for all $P \in \cP_o^n$ and $\phi \in \VL[n]$.

	\section{Proof of the Main Results}

\subsection{The $1$-dimensional case}
\input{proof-1-dim}

\subsection{The $2$-dimensional case}

\subsubsection{Some tensor equations}
\input{proof-2-dim-equations}

\subsubsection{Splitting over pyramids}
\input{proof-2-dim-pyramids}

\subsubsection{The main results} \label{sec: planar symmetric case}
\input{proof-2-dim-SL}

\subsection{The $n$-dimensional case}
\input{proof-n-dim}

	
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\bib{ZouXiong14}{article}{
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}

		\end{biblist}
	\end{bibdiv}

	
 \goodbreak \noindent

\begin{minipage}{0.63\textwidth}
	Christoph Haberl \\
	Vienna University of Technology \\
	Institute of Discrete Mathematics and Geometry \\
	Wiedner Hauptstraße 8--10/104 \\
	1040 Wien, Austria \\
	christoph.haberl@tuwien.ac.at
\end{minipage}
\begin{minipage}{0.35\textwidth}
	Lukas Parapatits \\
	ETH Zurich \\
	Department of Mathematics \\
	Rämistrasse 101 \\
	8092 Zürich, Switzerland \\
	lukas.parapatits@math.ethz.ch
\end{minipage}

\end{document}