#  $\mathrm{SL}(n)$ covariant function-valued valuations

Jin Li  Institut für Diskrete Mathematik und Geometrie, Technische Universität
Wien, Vienna, Austria
<EMAIL_ADDRESS>
###### Abstract

Classifications of $\mathrm{SL}(n)$ covariant function-valued valuations are
established with some assumptions of continuity. New valuations, for example,
weighted moment functions, are introduced and our classifications give unified
characterizations of the Laplace transform on convex bodies, $L_{p}$ moment
bodies, $L_{p}$ difference bodies, and polar $L_{p}$ moment bodies ($L_{-p}$
intersection bodies). Using the new classifications, we also establish some
Euler-type relations.

##  1 Introductions

Let $\mathcal{K}^{n}$ be the space of convex bodies (i.e., compact convex
sets) in $n$-dimensional Euclidean space ${\mathbb{R}}^{n}$. A valuation is a
map $Z$ from a subspace of $\mathcal{K}^{n}$ to an Abelian semigroup
$\langle\mathcal{A},+\rangle$ such that

$\displaystyle ZK+ZL=Z(K\cup L)+Z(K\cap L),$ (1)

whenever $K,L,K\cup L,K\cap L$ are contained in this subspace. Started with
the celebrated Hadwiger characterization theorem of intrinsic volumes, the
theory of valuations has attracted great interest, especially in the last
decades; see [1, 46, 2, 3, 5, 6, 10, 20, 19, 22, 25, 32, 33, 35, 47, 45, 23].
A valuation $Z$ taking values in a function space is a function-valued
valuation if the addition in (1) is the ordinary addition of functions.

This paper aims to give classifications of $\mathrm{SL}(n)$ covariant
function-valued valuations, only with some assumptions of continuity.
Meanwhile, we find new valuations, for example, weighted moment functions. Our
classifications establish unified characterizations of the Laplace transform
on convex bodies [30]; $L_{p}$ moment bodies and $L_{p}$ difference bodies for
$p\geq 1$ ($L_{p}$ Minkowski valuations) [32, 17, 49, 43, 29]; and polar
$L_{p}$ moment bodies for $p>-1$ [16]. The polar $L_{p}$ moment bodies are
also called $L_{-p}$ intersection bodies which are characterized in [18] for
$0<-p<1$. Polar $L_{0}$ moment bodies [51] are not characterized in [16], but
they are also included in our classifications. Above body-valued valuations
are embedded in the space of function-valued valuations by their
representations: support functions (for convex bodies) and gauge functions
(for star bodies).

Let $F({\mathbb{R}}^{n})$ be the space of all functions
$f:{\mathbb{R}}^{n}\to{\mathbb{R}}$. The special linear group is denoted by
$\mathrm{SL}(n)$. A map $Z:\mathcal{K}^{n}\to F({\mathbb{R}}^{n})$ is
$\mathrm{SL}(n)$ covariant if

$\displaystyle Z(\phi K)(x)=ZK(\phi^{t}x)$

for any $\phi\in\mathrm{SL}(n)$ and $x\in{\mathbb{R}}^{n}$, where $\phi^{t}$
is the transpose of $\phi$. If $Z:\mathcal{K}^{n}\to F({\mathbb{R}}^{n})$ is
an $\mathrm{SL}(n)$ covariant valuation, then the map $K\mapsto ZK(o)$ is an
$\mathrm{SL}(n)$ invariant real-valued valuation. Classifications of
$\mathrm{SL}(n)$ invariant real-valued valuations were known before; see, e.g.
[36] (or Corollary 35). Hence we only deal with $\mathrm{SL}(n)$ covariant
valuations with values in $F({\mathbb{R}}^{n}\setminus\\{o\\})$: the space of
all functions $f:{\mathbb{R}}^{n}\setminus\\{o\\}\to{\mathbb{R}}$.

Let $\mathcal{P}_{o}^{n}$ be the space of polytopes containing the origin. A
signed Radon measure is called continuous if the measure of every singleton is
$0$. Equivalently, the (Lebesgue) decomposition of this measure only gives an
absolutely continuous part and a singularly continuous part. We denote by
$\mathcal{M}^{c}({\mathbb{R}})$ the space of signed and continuous Radon
measures on ${\mathbb{R}}$. Let $Z$ map a subspace of $\mathcal{K}^{n}$ to
$F({\mathbb{R}}^{n}\setminus\\{o\\})$. We call $Z$ continuous if $ZK_{i}(x)\to
ZK(x)$ for every $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$ whenever $K_{i},K$ are
contained in this subspace and $K_{i}\to K$ with respect to the Hausdorff
metric.

######  Theorem 1.

Let $n\geq 3$. A map $Z:\mathcal{P}_{o}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ is a continuous and $\mathrm{SL}(n)$
covariant valuation if and only if there are $\zeta\in C({\mathbb{R}})$ and
$\mu\in\mathcal{M}^{c}({\mathbb{R}})$ such that

$\displaystyle ZP(x)$
$\displaystyle=\zeta(h_{P}(x))+\zeta(-h_{-P}(x))+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(P\cap
H_{x,t})d\mu(t)$

for every $P\in\mathcal{P}_{o}^{n}$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

Here $h_{P}$ is the support function of $P$, and $V_{n-1}(P\cap H_{x,t})$ is
the section volume (that is, the $(n-1)$-dimensional volume) of $P\cap
H_{x,t}$, where $H_{x,t}=\\{y\in{\mathbb{R}}^{n}:x\cdot y=t\\}$. The function
$h_{P}^{p}+h_{-P}^{p}$ is the $p$-th power of the support function of $L_{p}$
difference body. If $\mu$ has a locally integrable density $\zeta\in
F({\mathbb{R}})$, by Fubini’s theorem,

$\displaystyle\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(K\cap
H_{x,t})\zeta(t)dt=\int_{K}\zeta(x\cdot y)dy=:M_{\zeta}K(x)$ (2)

for any $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$ and $K\in\mathcal{K}^{n}$. If
$\zeta(t)=e^{-t}$ for $t\in{\mathbb{R}}$, then $M_{\zeta}K$ is the Laplace
transform of $K$. If $\zeta(t)=|t|^{p}$, then $M_{\zeta}K^{1/p}$ is the
support function of the $L_{p}$ moment body for $p\geq 1$ and the gauge
function of the polar $L_{p}$ moment body ($L_{-p}$ intersection body) for
$p>-1$, $p\neq 0$. If $\zeta(t)=\log|t|$ for $t\neq 0$ and $0$ for $t=0$, then
$\log M_{\zeta}K$ is the gauge function of the polar $L_{0}$ moment body. We
call $M_{\zeta}K$ the weighted moment function of $K$. See §3 for details.

Removing the continuity of $Z$ in Theorem 1, we obtain additional valuations
even for valuations with values in $C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$:
the space of continuous functions $f$ on ${\mathbb{R}}^{n}\setminus\\{o\\}$
such that $\lim_{r\to 0}rf(rx)=0$ for any
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$. For a function $\zeta$ on
${\mathbb{R}}$, let $\zeta^{R}$ denote its reflection, that is,
$\zeta^{R}(r)=\zeta(-r)$ for any $r\in{\mathbb{R}}^{n}$. Let $\mathcal{P}^{n}$
denote the space of all polytopes in ${\mathbb{R}}^{n}$. We say a valuation
$Z:\mathcal{P}^{n}\to C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$ is regular if,
for any $P\in\mathcal{P}^{n}$ and $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$, the
function $s\mapsto Z(sP)(s^{-1}x)$ is regular: bounded or measurable on any
compact interval $I\subset(0,\infty)$. All assumptions of the following
theorem are fulfilled for continuous and $\mathrm{SL}(n)$ covariant
$F({\mathbb{R}}^{n}\setminus\\{o\\})$-valued valuations. Also, $f\in
C({\mathbb{R}}^{n})$ implies $\lim_{r\to 0}rf(ru)=0$ for any $u\in S^{n-1}$.

######  Theorem 2.

Let $n\geq 3$. A map $Z:\mathcal{P}_{o}^{n}\to
C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$ is a regular and $\mathrm{SL}(n)$
covariant valuation if and only if there are $\zeta_{1},\zeta_{2}\in
C({\mathbb{R}})$ and $\mu\in\mathcal{M}^{c}({\mathbb{R}})$ such that

$\displaystyle ZP(x)$
$\displaystyle=\zeta_{1}(h_{P}(x))+\zeta_{1}^{R}(h_{-P}(x))+E^{-}_{\zeta_{2}}(P)(x)+E^{-}_{\zeta_{2}^{R}}(-P)(x)+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(P\cap
H_{x,t})d\mu(t)$

for every $P\in\mathcal{P}_{o}^{n}$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

For $P\in\mathcal{P}_{o}^{n}$, the new valuation $E^{-}_{\zeta}P$ is defined
by

$E^{-}_{\zeta}P(x):=\sum_{F\in\mathcal{F}^{-}(P)}(-1)^{\dim
F}\zeta(h_{F}(x)),$

where $\dim F$ is the dimension of $F$ and $\mathcal{F}^{-}(P)$ is the class
of faces (including $P$ itself) of $P$ which contain the origin. It is related
to Euler-type relations of support functions (Corollary 24). For more
information on $E^{-}_{\zeta}$ and related functionals, see §3.

For certain discontinuous $\zeta$, the valuation $P\mapsto\zeta(h_{P})$ is
still important in convex geometry. One example is the characteristic function
of the polar body. Hence, we also characterize the valuations $\zeta(h_{P})$
and $E^{-}_{\zeta}(P)$ for all $\zeta\in F({\mathbb{R}})$; see §2 for details.

Dual theorems, that is, classifications of $\mathrm{SL}(n)$ contravariant
$C({\mathbb{R}}^{n})$-valued valuations, are established in [28].
$\mathrm{SL}(n)$ contravariant $C({\mathbb{R}}^{n})$-valued valuations are
related to Orlicz mixed volumes introduced in [13, 50]. Our classification of
$\mathrm{SL}(n)$ covariant function-valued valuations might also contribute to
the recently developing Orlicz Brunn-Minkowski theory and the $L_{p}$ Brunn-
Minkowski theory for $p<1$ [40, 39, 7, 8, 35, 13, 50].

Recently, many interesting theorems for valuations on functions were
established by extending the corresponding theorems for valuations on convex
bodies; e.g. [11, 34, 4, 10, 30, 9, 42]. Our results will also be used to give
a characterization of the Legendre transform and the Laplace transform on
convex functions; see [27].

The paper is organized as follows. In §2, we give a classification of
valuations which are $\mathrm{GL}_{+}(n)$ (or $\mathrm{GL}(n)$) covariance of
weight $0$ so that the assumptions of continuity or regularity in Theorem 1
and Theorem 2 can be removed. Notation and background information are
collected in §3. We also give additional examples of valuations that do not
take values at continuous functions. In §4, we prove the “necessarity” of our
theorems, that is, the “if” part, that those operators are valuations which
satisfy the desired properties, e.g., $\mathrm{SL}(n)$ covariance, continuity.
The sections 5-8 are contributed to the proofs of the “sufficiency”, the “only
if” part. Using our classifications, we establish new Euler-type relations and
give new proofs for Euler-type relations introduced by Shephard [48] and for
local Euler-type relations introduced by Kabluchko, Last, and Zaporozhets
[24]; see §9. Classifications of valuations on $\mathcal{P}^{n}$ and
$\mathcal{K}^{n}$ are given in §10. We pick up some explicit characterizations
of some operators as applications of our theorems in the last section. Some of
them are known before and some of them are new.

##  2 $\mathrm{GL}_{+}(n)$ covariant valuations of weight $0$

The general linear group is denoted by $\mathrm{GL}(n)$ and the group of non-
degenerate linear transforms with positive determinants by
$\mathrm{GL}_{+}(n)$. A map $Z:\mathcal{K}^{n}\rightarrow F({\mathbb{R}}^{n})$
is $\mathrm{GL}_{+}(n)$ (or $\mathrm{GL}(n)$) covariant of weight $0$ if

$\displaystyle Z(\phi K)(x)=ZK(\phi^{t}x)$

for any $\phi\in\mathrm{GL}_{+}(n)$ (or $\mathrm{GL}(n)$, respectively) and
$x\in{\mathbb{R}}^{n}$. If $\zeta\in F({\mathbb{R}})$ is not continuous, the
functions $E^{-}_{\zeta}P$ and $\zeta(h_{P})$ are not continuous in general.
We can characterize them by assuming that they are $\mathrm{GL}_{+}(n)$
covariant of weight $0$.

######  Theorem 3.

Let $n\geq 3$. A map $Z:\mathcal{P}_{o}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ is a $\mathrm{GL}_{+}(n)$ covariant
valuation of weight $0$ if and only if there are $\zeta_{1},\zeta_{2}\in
F({\mathbb{R}})$ such that

$\displaystyle ZP(x)=$
$\displaystyle\zeta_{1}(h_{P}(x))+\zeta_{1}^{R}(h_{-P}(x))+E^{-}_{\zeta_{2}}(P)(x)+E^{-}_{\zeta_{2}^{R}}(-P)(x)$
(3)

for every $P\in\mathcal{P}_{o}^{n}$ and $x\in{\mathbb{R}}^{n}$.

A map $Z:\mathcal{P}_{o}^{2}\to F({\mathbb{R}}^{2}\setminus\\{o\\})$ is a
$\mathrm{GL}(2)$ covariant valuation of weight $0$ if and only if (3) holds
for $n=2$.

Let $\mathbbm{1}_{K}(x)$ denote the characteristic function of a set
$K\subset{\mathbb{R}}^{n}$, that is, if $x\in K$, then $\mathbbm{1}_{K}(x)=1$,
otherwise $\mathbbm{1}_{K}(x)=0$. Notice that the characteristic functions of
polar bodies are included in the functions classified in Theorem 3 since
$\mathbbm{1}_{P^{\ast}}(x)=\zeta(h_{P}(x))=\zeta(\|x\|_{P^{\ast}})$ for
$\zeta(t)=\mathbbm{1}_{[0,1]}(t)$. Here
$P^{\ast}=\\{y\in{\mathbb{R}}^{n}:x\cdot y\leq 1\\}$ and
$\|x\|_{P^{\ast}}=\min\\{\lambda>0:x\in\lambda K\\}$ is the gauge function.
Also, the characteristic function of the normal cone of $P$ at the origin,
$N(P,o)$, is included in the functions classified in Theorem 3 by
$\mathbbm{1}_{N(P,o)}(x)=\zeta(h_{P}(x))$ for $\zeta(t)=\mathbbm{1}_{0}(t)$.

##  3 Preliminaries and Notation

We refer to Schneider [44] as a general reference for convex geometry and
valuations theory.

Let $x\cdot y$ denote the inner product of $x,y\in{\mathbb{R}}^{n}$. The
Euclidean norm of $x\in{\mathbb{R}}^{n}$ is denoted by $|x|$. The support
function of a convex body $K$ is

$\displaystyle h_{K}(x)=\max_{y\in K}\\{x\cdot
y\\},~{}~{}x\in{\mathbb{R}}^{n}.$

The Hausdorff distance of $K,L\in\mathcal{K}^{n}$ is $\delta(K,L)=\max_{u\in
S^{n-1}}|h_{K}(u)-h_{L}(u)|$. The gauge function $\|x\|_{K^{\ast}}=h_{K}(x)$
if $K\in\mathcal{K}^{n}$ contains the origin.

For $p\geq 1$, the $L_{p}$ moment body $M_{p}K$ of a convex body $K$
containing the origin, is defined by

$\displaystyle h_{M_{p}K}(x)^{p}=\int_{K}|x\cdot
y|^{p}dy,~{}~{}x\in{\mathbb{R}}^{n}.$ (4)

The $L_{p}$ moment body is a fundamental operator in convex geometry. For the
related $L_{p}$ affine isoperimetric inequality, see [37, 38, 21]. It is
called an $L_{p}$ Minkowski valuation since it is a valuation with respect to
the $L_{p}$ Minkowski addition $h_{K+_{p}L}^{p}:=h_{K}^{p}+h_{L}^{p}$. The
volume-normalized $L_{p}$ moment bodies are call $L_{p}$ centroid bodies.

For $-1<p<1$, the function $(\int_{K}|x\cdot y|^{p}dy)^{1/p}$ is not a support
function anymore. But the polar $L_{p}$ moment body
$M_{p}^{\ast}K=(M_{p}K)^{\ast}$, a star-shaped body, is still interesting. The
gauge function of $M_{p}^{\ast}K$ is defined by

$\displaystyle\|x\|_{M_{p}^{\ast}K}^{p}=\int_{K}|x\cdot
y|^{p}dy,~{}~{}x\in{\mathbb{R}}^{n}.$ (5)

for $p>-1$, $p\neq 0$, and

$\displaystyle V(K)\log\|x\|_{M_{0}^{\ast}K}=\int_{K}\log|x\cdot
y|dy,~{}~{}x\in{\mathbb{R}}^{n}.$

for $p=0$. Polar $L_{-p}$ moment bodies are also called $L_{p}$ intersection
bodies since

$\lim_{p\uparrow 1}\Gamma(1-p)\|x\|_{M_{-p}^{\ast}K}^{-p}=\|x\|_{IK}^{-1},$

where $\Gamma$ denotes the Gamma function; see [15]. Here $IK$ is the
intersection body of $K$ which is defined by

$\displaystyle\|x\|_{IK}^{-1}=\frac{1}{|x|}V_{n-1}(K\cap H_{x,0}).$

$L_{p}$ Busemann-Petty problems are studied in [51, 14].

The Laplace transform $\mathcal{L}K$ of a convex body $K$ is

$\displaystyle\mathcal{L}K(x)=\int_{K}e^{-x\cdot
y}dy,~{}~{}x\in{\mathbb{R}}^{n}.$ (6)

Making use of the logarithmic version of this transform, Klartag [26] found so
far the best estimate on Bourgain’s slicing problem, which is one of the main
open problems in asymptotic geometry. Also, the characterization of the
Laplace transform on convex bodies is fundamental in the characterization of
the classic Laplace transform on Lebesgue functions; see [30].

Comparing (4) with (5) and (6), we are led to define the weighted moment
function

$\displaystyle M_{\zeta}K(x):=\int_{K}\zeta(x\cdot
y)dy,~{}~{}x\in{\mathbb{R}}^{n},K\in\mathcal{K}^{n}$

for a locally integrable function $\zeta:{\mathbb{R}}\to{\mathbb{R}}$.
Inspired by (2), we can further define the more general valuation:

$\displaystyle M_{\mu}(K)(x)=\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(K\cap
H_{x,t})d\mu(t),~{}~{}x\in{\mathbb{R}}^{n}\setminus\\{o\\}$

where $\mu$ is a signed Radon measure. If $\mu$ is the Dirac-delta measure
concentrated at $0$, then $M_{\mu}(K)=\|x\|_{IK}^{-1}$, which is characterized
by Ludwig [33].

We only characterize $M_{\mu}K$ for continuous measures since it is not
continuous if $\mu$ is a discrete measure.

Another $\mathrm{SL}(n)$ covariant valuation not included in our theorems is

$K\mapsto\zeta(h_{K}(x))V\left(K,x\right),$

where $V\left(K,x\right)$ is the volume of the cone $\left\\{\lambda
y\in{\mathbb{R}}^{n}:\lambda\in[0,1],~{}y\cdot x=h_{K}\left(x\right)\right\\}$
and $\zeta\in F({\mathbb{R}})$. For some homogeneous $\zeta$, this valuation
is characterized by Haberl and Parapatits [20] as an $\mathrm{SL}(n)$
covariant valuation valued in a special function space, and is used to
characterize $L_{p}$ surface area measures.

Let $P\in\mathcal{P}^{n}$. We say a set $F$ is a face of $P$ if either $F=P$
or $F=P\cap H_{u,h_{P}(u)}$ for some $u\in S^{n-1}$. The normal cone
$N(P,F)=\\{u\in{\mathbb{R}}^{n}\setminus\\{o\\}:F\subseteq P\cap
H_{u,h_{P}(u)}\\}\cup\\{o\\}$. We denote by $\mathcal{F}(P)$ the class of
faces of $P$, and by $\mathcal{F}^{-}(P)$ ($\mathcal{F}^{+}(P)$) the class of
faces $F$ of $P$ such that $h_{P}(u)\leq 0$ ($h_{P}(u)\geq 0$, respectively)
for all $u\in N(P,F)$. Let $\dim K$ be the dimension of $K\in\mathcal{K}^{n}$,
i.e., the dimension of the affine hull of $K$. For any $\zeta\in
F({\mathbb{R}})$, we define the maps
$E^{\pm}_{\zeta},E_{\zeta}:\mathcal{P}^{n}\to F({\mathbb{R}}^{n})$ by

$\displaystyle E^{\pm}_{\zeta}P(x):=\sum_{F\in\mathcal{F}^{\pm}(P)}(-1)^{\dim
F}\zeta(h_{F}(x)),$ (7) $\displaystyle
E_{\zeta}P(x):=\sum_{F\in\mathcal{F}(P)}(-1)^{\dim F}\zeta(h_{F}(x))$ (8)

for any $P\in\mathcal{P}^{n}$ and $x\in{\mathbb{R}}^{n}$. If $o\in P$, then
$\mathcal{F}^{+}(P)=\mathcal{F}(P)$ and $\mathcal{F}^{-}(P)$ is the class of
faces of $P$ which contain the origin.

For $0\leq j\leq\dim P$, let $\mathcal{F}_{j}(P)=\\{F\in\mathcal{F}(P):\dim
F=j\\}$ and $\mathcal{F}^{\pm}_{j}(P)=\\{F\in\mathcal{F}^{\pm}(P):\dim F=j\\}$
Also, let $|\mathcal{F}_{j}(P)|$ and $|\mathcal{F}^{\pm}_{j}(P)|$ be their
cardinalities.

The convex hull of a set $A$ is denoted by $[A]$ and the convex hull of a set
$A$ and a point $x\in{\mathbb{R}}^{n}$ will be briefly written as $[A,x]$
instead of $[A,\\{x\\}]$.

##  4 Properties of the classified valuations

We call a valuation simple if it vanishes on lower-dimensional sets. Let
$C^{\infty}({\mathbb{R}}^{n})$ be the space of infinitely differentiable
functions on ${\mathbb{R}}^{n}$. Denote by
$AC_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$ the space of continuous functions
$f:{\mathbb{R}}^{n}\setminus\\{o\\}\to{\mathbb{R}}$ satisfying that, for any
$u\in S^{n-1}$,
1\. the function $r\mapsto rf(ru)$ is absolutely continuous on $(0,s]$ for any
$s>0$;
2\. $\lim_{r\to 0}rf(ru)=0$ for any $u\in S^{n-1}$.

######  Lemma 4.

Let $\mu\in\mathcal{M}^{c}({\mathbb{R}})$. The map $Z$ defined on
$\mathcal{K}^{n}$ by

$\displaystyle ZK(x)=\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(K\cap
H_{x,t})d\mu(t),~{}~{}K\in\mathcal{K}^{n},~{}~{}x\in{\mathbb{R}}^{n}\setminus\\{o\\},$
(9)

is a continuous, simple, $\mathrm{SL}(n)$ covariant
$C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$-valued valuation.

Moreover, if $\mu$ has a density $\zeta\in F({\mathbb{R}})$, then

$\displaystyle ZK(x)=\int_{K}\zeta(x\cdot y)dy,$ (10)

and $ZK\in AC_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$ for every
$K\in\mathcal{K}^{n}$. Also, if $\zeta\in C^{\infty}({\mathbb{R}})$, (10) can
be defined also at $x=o$ and $ZK\in C^{\infty}({\mathbb{R}}^{n})$ for every
$K\in\mathcal{K}^{n}$.

###### Proof.

The simplicity and the valuation property of $Z$ are trivial.

Since the valuation defined by (10) is clearly $\mathrm{SL}(n)$ covariant, by
(2) and a standard approximation, the valuation defined by (9) is also
$\mathrm{SL}(n)$ covariant.

Let $\\{x_{i}\\}\subset{\mathbb{R}}^{n}\setminus\\{o\\}$ be a sequence
converging to $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$ and
$\\{K_{i}\\}\subset\mathcal{K}^{n}$ be a sequence converging to
$K\in\mathcal{K}^{n}$. By [44, Theorem 1.8.8], $K_{i}\cap H_{x_{i},t}\to K\cap
H_{x,t}$ if either $H_{x,t}$ passes through the relative interior of $K$ or
$K\cap H_{x,t}=\emptyset$. Hence $V_{n-1}(K_{i}\cap H_{x_{i},t})\to
V_{n-1}(K\cap H_{x,t})$ for every
$t\in{\mathbb{R}}\setminus\\{h_{K}(x),-h_{K}(-x)\\}$. Notice that the support
set of the function $t\mapsto V_{n-1}(K\cap H_{x,t})$ is compact. By the
dominated convergence theorem, (9), the boundedness of $t\mapsto V_{n-1}(K\cap
H_{x,t})$ and the continuity of $\mu$, we obtain $ZK_{i}(x_{i})\to ZK(x)$.
This not only gives the continuity of $ZK$, but also the continuity of $Z$.

For $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$, let $r=|x|$ and $u=x/r$. By (9),

$\displaystyle rZK(ru)=\int_{-rh_{-K}(u)}^{rh_{K}(u)}V_{n-1}(K\cap
H_{ru,t})d\mu(t).$

Since $\mu$ is continuous and

$V_{n-1}(K\cap H_{ru,t})\leq\max_{t\in{\mathbb{R}}}V_{n-1}(K\cap H_{u,t}),$

we have $\lim_{r\to 0}rZK(ru)=0$.

If $\mu$ has a density $\zeta$, then $\zeta$ is locally integrable. For any
$\epsilon>0$, since $|\zeta(t)V_{n-1}(K\cap
H_{ru,t})|\leq|\zeta(t)|\max_{t\in{\mathbb{R}}}V_{n-1}(K\cap H_{u,t})$ and
$\zeta$ is locally integrable, there is a $\delta>0$ such that

$\displaystyle\left|\int_{E}\zeta(t)V_{n-1}(K\cap
H_{ru,t})dt\right|<\epsilon,$

whenever $|E|<\delta$ for compact $E\subset{\mathbb{R}}$. Hence the function

$\displaystyle r\mapsto rZK(ru)=\int_{0}^{rh_{K}(u)}\zeta(t)V_{n-1}(K\cap
H_{ru,t})d\mu(t)+\int_{-rh_{-K}(u)}^{0}\zeta(t)V_{n-1}(K\cap H_{ru,t})d\mu(t)$

is absolutely continuous on $(0,s]$ for any $s>0$.

If further $\zeta\in C^{\infty}({\mathbb{R}})$, then clearly $ZK\in
C^{\infty}({\mathbb{R}}^{n})$ for any $K\in\mathcal{K}^{n}$. ∎

Since

$\displaystyle h_{K\cup L}=\max\\{h_{K},h_{L}\\},h_{K\cap
L}=\min\\{h_{K},h_{L}\\}$

for $K,L,K\cup L\in\mathcal{K}^{n}$, the map $K\mapsto\zeta(h_{K})$ is a
valuation. The following two lemmas are trivial, so we omit the proofs.

######  Lemma 5.

Let $\zeta\in F({\mathbb{R}}^{n})$. The map $Z:\mathcal{K}^{n}\to
F({\mathbb{R}}^{n})$, defined by $ZK=\zeta(h_{K})$, is a $\mathrm{GL}(n)$
covariant valuation of weight $0$. Moreover, if $\zeta\in C({\mathbb{R}})$,
then $ZK\in C({\mathbb{R}}^{n})$ and $Z$ is continuous.

######  Lemma 6.

If $Z:\mathcal{K}^{n}\to F({\mathbb{R}}^{n})$ is a continuous,
$\mathrm{GL}(n)$ covariant valuation of weight $0$, so are
$Z^{\prime},Z^{\prime\prime}$ defined by $Z^{\prime}K=Z(-K)$ and
$Z^{\prime\prime}K=Z[K,o]$.

To check that $E^{\pm}_{\zeta}$, $E_{\zeta}$ are valuations on
$\mathcal{P}^{n}$, we only need to check that (1) holds for some special
$K,L\in\mathcal{P}_{o}^{n}$.

######  Lemma 7 ([44], Theorem 6.2.3).

If a map $Z:\mathcal{P}^{n}\to{\mathbb{R}}$ satisfies that

$\displaystyle Z(P)+Z(P\cap H)=Z(P\cap H^{+})+Z(P\cap H^{-})$ (11)

for any $P\in\mathcal{P}^{n}$ and hyperplane $H$, then $Z$ is a valuation.
Here $H^{+},H^{-}$ denote the half-spaces bounded by $H$.

######  Lemma 8.

The operators $E^{\pm}_{\zeta}:\mathcal{P}^{n}\to F({\mathbb{R}}^{n})$ and
$E_{\zeta}:\mathcal{P}^{n}\to F({\mathbb{R}}^{n})$ are $\mathrm{GL}(n)$
covariant valuations of weight $0$. Moreover, if $\zeta\in C({\mathbb{R}})$,
then $E^{\pm}_{\zeta}(P)\in C({\mathbb{R}}^{n})$ for any
$P\in\mathcal{P}^{n}$.

###### Proof.

By the definitions (7) and (8) together with Lemma 5, $E^{\pm}_{\zeta}$ and
$E_{\zeta}$ are $\mathrm{GL}(n)$ covariant of weight $0$ and the last
statement holds.

In the following, we only show that $E^{-}_{\zeta}$ is a valuation. The
argument for $E^{+}_{\zeta}$ and $E_{\zeta}$ are similar. By Lemma 7, we only
need to check that (11) holds for any $P\in\mathcal{P}^{n}$ and hyperplane
$H$. We can assume that $\dim(P\cap H)<\dim P=\dim(P\cap H^{+})=\dim(P\cap
H^{-})$. We can also assume that $\dim P=n$ since otherwise, we can consider
the affine hull of $P$ for the following argument. Now we have $\dim(P\cap
H)=n-1$.

Let $0\leq j\leq n$. We can separate $\mathcal{F}_{j}(P)$ as follows

$\displaystyle\mathcal{F}_{j}(P)=\mathcal{A}_{j,H^{+}}\cup\mathcal{A}_{j,H^{-}}\cup\mathcal{B}_{j}\cup\mathcal{C}_{j},$
(12)

where

$\displaystyle\mathcal{A}_{j,H^{\pm}}=\\{F\in\mathcal{F}_{j}(P):\operatorname{relint}F\subset\operatorname{relint}H^{\pm}\\},$
$\displaystyle\mathcal{B}_{j}=\\{F\in\mathcal{F}_{j}(P):F\subset H\\},$
$\displaystyle\mathcal{C}_{j}=\\{F\in\mathcal{F}_{j}(P):F\cap\operatorname{relint}H^{+}\neq\emptyset,F\cap\operatorname{relint}H^{-}\neq\emptyset\\}.$

Set $\mathcal{C}_{n+1}=\emptyset$ and notice that $\mathcal{C}_{n}=\\{P\\}$.
Since $\partial(P\cap H)=\partial P\cap H$ and $\partial(P\cap
H^{\pm})=(\partial P\cap H^{\pm})\cup(P\cap H)$, it is easy to see

$\displaystyle\mathcal{F}_{j}(P\cap H)=\mathcal{B}_{j}\cup\\{F\cap
H:F\in\mathcal{C}_{j+1}\\},$ (13)

and

$\displaystyle\mathcal{F}_{j}(P\cap
H^{+})=\mathcal{A}_{j,H^{+}}\cup\mathcal{B}_{j}\cup\\{F\cap
H^{+}:F\in\mathcal{C}_{j}\\}\cup\\{F\cap H:F\in\mathcal{C}_{j+1}\\},$ (14)
$\displaystyle\mathcal{F}_{j}(P\cap
H^{-})=\mathcal{A}_{j,H^{-}}\cup\mathcal{B}_{j}\cup\\{F\cap
H^{-}:F\in\mathcal{C}_{j}\\}\cup\\{F\cap H:F\in\mathcal{C}_{j+1}\\}.$ (15)

Let
$\mathcal{A}_{j,H^{\pm}}^{-}=\mathcal{A}_{j,H^{\pm}}\cap\mathcal{F}^{-}_{j}(P)$,
$\mathcal{B}_{j}^{-}=\mathcal{B}_{j}\cap\mathcal{F}^{-}_{j}(P)$ and
$\mathcal{C}_{j}^{-}=\mathcal{C}_{j}\cap\mathcal{F}^{-}_{j}(P)$.

For the case $o\in H$, (12)-(15) also hold if we replace
$\mathcal{F},\mathcal{A}_{j,H^{\pm}},\mathcal{B}_{j},\mathcal{C}_{j}$ by
$\mathcal{F}^{-},\mathcal{A}_{j,H^{\pm}}^{-},\mathcal{B}_{j}^{-},\mathcal{C}_{j}^{-}$,
respectively. Now combined with the definition (7), we have

$\displaystyle E^{-}_{\zeta}P=\sum_{0\leq j\leq
n}(-1)^{j}\left(\sum_{F\in\mathcal{A}_{j,H^{+}}^{-}}\zeta(h_{F})+\sum_{F\in\mathcal{A}_{j,H^{-}}^{-}}\zeta(h_{F})+\sum_{F\in\mathcal{B}_{j}^{-}}\zeta(h_{F})+\sum_{F\in\mathcal{C}_{j}^{-}}\zeta(h_{F})\right),$
$\displaystyle E^{-}_{\zeta}(P\cap H)=\sum_{0\leq j\leq
n-1}(-1)^{j}\left(\sum_{F\in\mathcal{B}_{j}^{-}}\zeta(h_{F})+\sum_{F\in\mathcal{C}_{j+1}^{-}}\zeta(h_{F\cap
H})\right)$ $\displaystyle E^{-}_{\zeta}(P\cap H^{+})=\sum_{0\leq j\leq
n}(-1)^{j}\left(\sum_{F\in\mathcal{A}_{j,H^{+}}^{-}}\zeta(h_{F})+\sum_{F\in\mathcal{B}_{j}^{-}}\zeta(h_{F})+\sum_{F\in\mathcal{C}_{j}^{-}}\zeta(h_{F\cap
H^{+}})+\sum_{F\in\mathcal{C}_{j+1}^{-}}\zeta(h_{F\cap H})\right),$
$\displaystyle E^{-}_{\zeta}(P\cap H^{-})=\sum_{0\leq j\leq
n}(-1)^{j}\left(\sum_{F\in\mathcal{A}_{j,H^{-}}^{-}}\zeta(h_{F})+\sum_{F\in\mathcal{B}_{j}^{-}}\zeta(h_{F})+\sum_{F\in\mathcal{C}_{j}^{-}}\zeta(h_{F\cap
H^{-}})+\sum_{F\in\mathcal{C}_{j+1}^{-}}\zeta(h_{F\cap H})\right).$

Notice that $\mathcal{B}_{n}^{-}=\emptyset$. To check (11) holds for
$Z=E^{-}_{\zeta}$, after eliminating the same items, we only need to show

$\displaystyle\sum_{0\leq j\leq
n}(-1)^{j}\sum_{F\in\mathcal{C}_{j}^{-}}\zeta(h_{F})+\sum_{0\leq j\leq
n-1}(-1)^{j}\sum_{F\in\mathcal{C}_{j+1}^{-}}\zeta(h_{F\cap H})$
$\displaystyle=\sum_{0\leq j\leq
n}(-1)^{j}\sum_{F\in\mathcal{C}_{j}^{-}}\zeta(h_{F\cap H^{+}})+\sum_{0\leq
j\leq n}(-1)^{j}\sum_{F\in\mathcal{C}_{j}^{-}}\zeta(h_{F\cap
H^{-}})+2\sum_{0\leq j\leq
n-1}(-1)^{j}\sum_{F\in\mathcal{C}_{j+1}^{-}}\zeta(h_{F\cap H}).$ (16)

The equation (4) is actually true since

$\displaystyle\zeta(h_{F})+\zeta(h_{F\cap H})$ $\displaystyle=\zeta(h_{F\cap
H^{+}})+\zeta(h_{F\cap H^{-}}),$

and $\mathcal{C}_{0}^{-}=\emptyset$ (by its definition).

For the case $o\notin H$,

$\displaystyle\mathcal{F}^{-}_{j}(P)=\mathcal{A}_{j,H^{+}}^{-}\cup\mathcal{A}_{j,H^{-}}^{-}\cup\mathcal{B}_{j}^{-}\cup\mathcal{C}_{j}^{-},$
(17)

and

$\displaystyle\mathcal{F}^{-}_{j}(P\cap H)=\emptyset.$ (18)

Set
$\mathcal{B}_{j,H^{\pm}}^{-}=\mathcal{B}_{j}^{-}\cap\mathcal{F}^{-}_{j}(P\cap
H^{\pm})$ and $\mathcal{D}_{j}^{-}=\\{F\cap H:F\in\mathcal{C}_{j+1}^{-}\\}$,
$\mathcal{D}_{j,H^{\pm}}^{-}=\mathcal{D}_{j}^{-}\cap\mathcal{F}^{-}_{j}(P\cap
H^{\pm})$. We have

$\displaystyle\mathcal{F}^{-}_{j}(P\cap
H^{+})=\mathcal{A}_{j,H^{+}}^{-}\cup\mathcal{B}_{j,H^{+}}^{-}\cup\\{F\cap
H^{+}:F\in\mathcal{C}_{j}^{-}\\}\cup\mathcal{D}_{j,H^{+}}^{-},$ (19)
$\displaystyle\mathcal{F}^{-}_{j}(P\cap
H^{-})=\mathcal{A}_{j,H^{-}}^{-}\cup\mathcal{B}_{j,H^{-}}^{-}\cup\\{F\cap
H^{-}:F\in\mathcal{C}_{j}^{-}\\}\cup\mathcal{D}_{j,H^{-}}^{-},$ (20)

and

$\displaystyle\mathcal{B}_{j,H^{+}}^{-}\cup\mathcal{B}_{j,H^{-}}^{-}=\mathcal{B}_{j}^{-},~{}~{}~{}\mathcal{B}_{j,H^{+}}^{-}\cap\mathcal{B}_{j,H^{-}}^{-}=\emptyset,$
(21)
$\displaystyle\mathcal{D}_{j,H^{+}}^{-}\cup\mathcal{D}_{j,H^{-}}^{-}=\mathcal{D}_{j}^{-},~{}~{}\mathcal{D}_{j,H^{+}}^{-}\cap\mathcal{D}_{j,H^{-}}^{-}=\emptyset.$
(22)

Combining (17)-(22) with the definition of $E^{-}_{\zeta}$, we only need to
show that

$\displaystyle\sum_{0\leq j\leq
n}(-1)^{j}\sum_{F\in\mathcal{C}_{j}^{-}}\zeta(h_{F})$
$\displaystyle=\sum_{0\leq j\leq
n}(-1)^{j}\sum_{F\in\mathcal{C}_{j}^{-}}\zeta(h_{F\cap H^{+}})+\sum_{0\leq
j\leq n}(-1)^{j}\sum_{F\in\mathcal{C}_{j}^{-}}\zeta(h_{F\cap
H^{-}})+\sum_{0\leq j\leq
n-1}(-1)^{j}\sum_{F\in\mathcal{C}_{j+1}^{-}}\zeta(h_{F\cap H}),$

which is the same as (4). ∎

Let $C({\mathbb{R}}^{n}\setminus\\{o\\})$ be the space of continuous functions
on ${\mathbb{R}}^{n}\setminus\\{o\\}$. We say a function
$f:(0,\infty)\to{\mathbb{R}}$ is regular if $f$ is bounded or measurable on
any compact interval.

######  Lemma 9.

If $Z:\mathcal{P}_{o}^{n}\to F({\mathbb{R}}^{n}\setminus\\{o\\})$ is
continuous and $\mathrm{SL}(n)$ covariant, then $ZP\in
C({\mathbb{R}}^{n}\setminus\\{o\\})$ for every $P\in\mathcal{P}_{o}^{n}$ and
the function $s\mapsto Z(sT^{n})(s^{-1}x)$ is regular on $(0,\infty)$ for any
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$. Moreover, if further $Z$ is simple and
$Z(sT^{n})(s^{-1}x)=s^{n}Z(T^{n})(x)$ for any $s>0$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$, then $\lim_{s\to 0}sZT^{n}(sx)=0$ for
any $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

If $Z:\mathcal{P}^{n}\to F({\mathbb{R}}^{n}\setminus\\{o\\})$ is continuous
and $\mathrm{SL}(n)$ covariant, then $ZP\in
C({\mathbb{R}}^{n}\setminus\\{o\\})$ for every $P\in\mathcal{P}^{n}$ and the
functions $s\mapsto Z(sT^{n})(s^{-1}x)$ and $s\mapsto
Z(s[e_{1},\dots,e_{n}])(s^{-1}x)$ are regular on $(0,\infty)$ for any
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$. Moreover, if further $Z$ vanishes on
all $P\in\mathcal{P}_{o}^{n}$ with $\dim P<n$, and
$Z(sT^{n})(s^{-1}x)=s^{n}Z(T^{n})(x)$ and
$Z(s[e_{1},\dots,e_{n}])(s^{-1}x)=s^{n}Z([e_{1},\dots,e_{n}])(x)$ for any
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$, then $\lim_{s\to 0}sZT^{n}(sx)=0$ and
$\lim_{s\to 0}sZ[e_{1},\dots,e_{n}](sx)=0$ for any
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

###### Proof.

We only prove the first part of the Lemma. The second part is similar.

Let $x_{i},x\in{\mathbb{R}}^{n}\setminus\\{o\\}$ such that $x_{i}\to x$. We
need to show that $ZP(x_{i})\to ZP(x)$. There are $\phi_{i}\in\mathrm{SL}(n)$
such that $x_{i}=\phi_{i}^{t}x$ and $(\phi_{i}P)\to P$ with respect to the
Hausdorff metric. Hence the continuity and the $\mathrm{SL}(n)$ covariance of
$Z$ give $ZP(x_{i})=Z(\phi_{i}P)(x)\to ZP(x)$.

Let $\phi_{s}\in\mathrm{SL}(n)$ such that $\phi_{s}x=s^{-1}x$ and
$\phi_{s}y=s^{1/(n-1)}y$ for any $y\in
x^{\bot}:=\\{y\in{\mathbb{R}}^{n}:x\cdot y=0\\}$. Clearly
$\phi_{s}^{t}=\phi_{s}$. The $\mathrm{SL}(n)$ covariance of $Z$ gives
$Z(sT^{n})(s^{-1}x)=Z(\phi_{s}sT^{n})(x)$. Hence, the continuity of $Z$
implies that $s\mapsto Z(sT^{n})(s^{-1}x)$ is continuous which is clearly
regular.

Lastly, let $s>0$ and $\psi\in\mathrm{GL}_{+}(n)$ such that $\phi x=sx$ and
$\phi y=y$ for any $y\in x^{\bot}$. By the $\mathrm{SL}(n)$ covariance of $Z$
and the assumption that $Z(sT^{n})(s^{-1}x)=s^{n}Z(T^{n})(x)$ for any
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$, we have $Z(\psi
T^{n})(x)=Z(s^{1/n}T^{n})(s^{-1/n}\psi^{t}x)=sZT^{n}(sx)$. Also, $\psi T^{n}$
converges to the orthogonal projection of $T^{n}$ to $x^{\bot}$ as $r\to 0$.
Together with the simplicity of $Z$, we get the desired property. ∎

##  5 Functional equations and uniqueness arguments

If a regular function $f:(0,\infty)\to{\mathbb{R}}$ satisfies the Cauchy
functional equation

$\displaystyle f(a+b)=f(a)+f(b),$

for any $a,b>0$, then

$\displaystyle f(r)=f(1)r$

for any $r>0$.

The following dissections of standard simplices $T^{d}:=[o,e_{1},\dots,e_{d}]$
are fundamental in our arguments.

For $0<\lambda<1$, the hyperplane $H_{\lambda}$ and half spaces
$H_{\lambda}^{-},H_{\lambda}^{+}$ are defined by

$\displaystyle
H_{\lambda}:=\\{x\in{\mathbb{R}}^{n}:x\cdot((1-\lambda)e_{1}-\lambda
e_{2})=0\\},$ $\displaystyle
H_{\lambda}^{-}:=\\{x\in{\mathbb{R}}^{n}:x\cdot((1-\lambda)e_{1}-\lambda
e_{2})\leq 0\\},$ $\displaystyle
H_{\lambda}^{+}:=\\{x\in{\mathbb{R}}^{n}:x\cdot((1-\lambda)e_{1}-\lambda
e_{2})\geq 0\\}.$

Let $Z:\mathcal{P}_{o}^{n}\to F({\mathbb{R}}^{n}\setminus\\{o\\})$ be an
$\mathrm{SL}(n)$ covariant valuation. We have

$\displaystyle Z(sT^{d})(x)+Z(sT^{d}\cap H_{\lambda})(x)=Z(sT^{d}\cap
H_{\lambda}^{-})(x)+Z(sT^{d}\cap
H_{\lambda}^{+})(x),~{}~{}x\in{\mathbb{R}}^{n}\setminus\\{o\\}$ (23)

for any $s>0$ and $2\leq d\leq n$. Also, let
$\widehat{T}^{d-1}:=[o,e_{1},e_{3},\dots,e_{d}]$ and
$\phi_{\lambda},\psi_{\lambda}\in\mathrm{GL}(n)$ such that

$\phi_{\lambda}e_{1}=\lambda
e_{1}+\left(1-\lambda\right)e_{2},~{}\phi_{\lambda}e_{i}=e_{i},~{}\text{for}~{}2\leq
i\leq n,$

and

$\psi_{\lambda}e_{2}=\lambda
e_{1}+\left(1-\lambda\right)e_{2},\psi_{\lambda}e_{i}=e_{i},~{}\text{for}~{}1\leq
i\leq n,i\neq 2.$

Notice that $sT^{d}\cap H_{\lambda}^{-}=\phi_{\lambda}sT^{d}$, $sT^{d}\cap
H_{\lambda}^{+}=\psi_{\lambda}sT^{d}$ and $sT^{d}\cap
H_{\lambda}=\phi_{\lambda}s\hat{T}^{d-1}$. Combined with the $\mathrm{SL}(n)$
covariance of $Z$, (23) implies

$\displaystyle
Z(sT^{d})(x)+Z(\lambda^{1/n}s\hat{T}^{d-1})(\lambda^{-1/n}\phi_{\lambda}^{t}x)$
$\displaystyle\qquad=Z(\lambda^{1/n}sT^{d})\left(\lambda^{-1/n}\phi_{\lambda}^{t}x\right)+Z((1-\lambda)^{1/n}sT^{d})\left((1-\lambda)^{-1/n}\psi_{\lambda}^{t}x\right),$
(24)

where $x=r_{1}e_{1}+\dots+r_{n}e_{n}\in{\mathbb{R}}^{n}\setminus\\{o\\}$,
$\phi_{\lambda}^{t}x=(\lambda
r_{1}+(1-\lambda)r_{2})e_{1}+r_{2}e_{2}+r_{3}e_{3}+\dots+r_{n}e_{n}$ and
$\psi_{\lambda}^{t}x=r_{1}e_{1}+(\lambda
r_{1}+(1-\lambda)r_{2})e_{2}+r_{3}e_{3}+\dots+r_{n}e_{n}$.

Now we establish some properties of $Z(sT^{d})(x)$ by assuming
$Z(s\hat{T}^{d-1})(x)=0$ in (5).

######  Lemma 10.

Let $d\geq 2$ and let the function
$f:(0,\infty)\times{\mathbb{R}}^{d}\setminus\\{o\\}\to{\mathbb{R}}$ satisfy
the following properties.
($i$)

$\displaystyle f\big{(}s^{1/n};y\big{)}$ $\displaystyle=f\big{(}(\lambda
s)^{1/n};\lambda^{-1/n}\phi_{\lambda}^{t}y\big{)}+f\big{(}((1-\lambda)s)^{1/n};(1-\lambda)^{-1/n}\psi_{\lambda}^{t}y\big{)}$
(25)

for any $s>0$, $\lambda\in(0,1)$ and $y\in{\mathbb{R}}^{d}\setminus\\{o\\}$.
($ii$) For any $s>0$, $y\in{\mathbb{R}}^{d}\setminus\\{o\\}$ and an arbitrary
permutation $\pi$ (even permutation for $d\geq 3$) of coordinates of $y$,

$\displaystyle f(s;y)=f(s;\pi y).$ (26)

We have following conclusions.
(a) If $f(s;re_{1}+\dots+re_{m})=0$ for any $s>0$, $r\neq 0$ and $1\leq m\leq
d$, then

$\displaystyle f(s;x)=0$ (27)

for every $s>0$ and $x\in{\mathbb{R}}^{d}\setminus\\{o\\}$. Moreover, if
$f(s;\cdot)$ is continuous, we only need the assumption $f(s;re_{1})=0$ to
conclude (27).
(b) For $d\geq 3$, if the function $s\mapsto
f(s^{1/n};s^{-1/n}r(e_{1}+\dots+e_{m}))$ is regular for any $r\neq 0$ and
$1\leq m\leq d$, then

$\displaystyle f(s^{1/n};s^{-1/n}x)=sf(1;x)$ (28)

for any $x\in{\mathbb{R}}^{d}\setminus\\{o\\}$.

For $d=2$, if the function $s\mapsto f(s^{1/n};s^{-1/n}r(e_{1}+e_{2}))$ is
regular for any $r\neq 0$, then

$\displaystyle f(s^{1/n};s^{-1/n}r(e_{1}+e_{2}))=sf(1;r(e_{1}+e_{2})).$

(c) For $d\geq 3$, if $f(s^{1/n};s^{-1/n}x)=f(1;x)$ for any $s>0$ and
$x\in{\mathbb{R}}^{d}\setminus\\{o\\}$, then (27) also holds. For $d=2$, if
additionally, $f(s;re_{1})=0$ for any $s>0$ and $r\neq 0$, then (27) also
holds.

###### Proof.

(a). We prove (27) by induction on $k$, the number of non-zero coordinates of
$x$. By (26), we only need to show that $f(s;x)=0$ for
$x=r_{1}e_{1}+\dots+r_{k}e_{k}$, where $r_{1},\dots r_{k}$ are non-zero
numbers. For $k=1$, it is trivial. Assume that the statement holds for $k-1$.
Set $\dot{x}=r_{3}e_{3}+\dots+r_{k}e_{k}$.

For $r_{1}>r_{2}>0$ or $0>r_{2}>r_{1}$, taking $\lambda=\frac{r_{2}}{r_{1}}$,
$y=(1-\lambda)^{1/n}(r_{1}e_{1}+\dot{x})$ and replacing $s$ with
$s/(1-\lambda)$ in (25), we get

$\displaystyle
f\left(\left(\frac{r_{1}}{r_{1}-r_{2}}s\right)^{1/n};\left(\frac{r_{1}-r_{2}}{r_{1}}\right)^{1/n}(r_{1}e_{1}+\dot{x})\right)$
$\displaystyle=f\left(\left(\frac{r_{2}}{r_{1}-r_{2}}s\right)^{1/n};\left(\frac{r_{1}-r_{2}}{r_{2}}\right)^{1/n}(r_{2}e_{1}+\dot{x})\right)+f\left(s^{1/n};r_{1}e_{1}+r_{2}e_{2}+\dot{x}\right).$
(29)

For $r_{2}>r_{1}>0$ or $0>r_{1}>r_{2}$, taking
$1-\lambda=\frac{r_{1}}{r_{2}}$, $y=\lambda^{1/n}(r_{2}e_{2}+\dot{x})$ and
replacing $s$ with $s/\lambda$ in (25), we get

$\displaystyle
f\left(\left(\frac{r_{2}}{r_{2}-r_{1}}s\right)^{1/n};\left(\frac{r_{2}-r_{1}}{r_{2}}\right)^{1/n}(r_{2}e_{2}+\dot{x})\right),$
$\displaystyle=f\left(s^{1/n};r_{1}e_{1}+r_{2}e_{2}+\dot{x}\right)+f\left(\left(\frac{r_{1}}{r_{2}-r_{1}}s\right)^{1/n};\left(\frac{r_{2}-r_{1}}{r_{1}}\right)^{1/n}(r_{1}e_{2}+\dot{x})\right).$
(30)

For $r_{1}>0>r_{2}$ or $r_{2}>0>r_{1}$, taking
$0<\lambda=\frac{r_{2}}{r_{2}-r_{1}}<1$ and $y=x$ in (25), we get

$\displaystyle f\left(s^{1/n};r_{1}e_{1}+r_{2}e_{2}+\dot{x}\right)$
$\displaystyle=$ $\displaystyle
f\left(\left(\frac{r_{2}}{r_{2}-r_{1}}s\right)^{1/n};\left(\frac{r_{2}-r_{1}}{r_{2}}\right)^{1/n}(r_{2}e_{2}+\dot{x})\right)+f\left(\left(\frac{-r_{1}}{r_{2}-r_{1}}s\right)^{1/n};\left(\frac{r_{2}-r_{1}}{-r_{1}}\right)^{1/n}(r_{1}e_{1}+\dot{x})\right).$
(31)

The induction assumption together with (5), (5) and (5) implies that
$f(s;x)=0$ for $x=r_{1}e_{1}+\dots+r_{k}e_{k}$ if $r_{1}\neq r_{2}$.

Now let $r_{1}=r_{2}=:r$. If $f(s;\cdot)$ is continuous, we get
$f(s;re_{1}+re_{2}+r_{3}e_{3}+\cdots+r_{k}e_{k})=0$ follows from the above
conclusions. Without the continuity of $f(s;\cdot)$, if all
$r_{1},\dots,r_{k}$ are the same, the case is what we assumed. Hence we can
say $r_{3}\neq r$. Also,
$f(s;re_{1}+re_{2}+r_{3}e_{3}+\dots+r_{k}e_{k})=f(s;re_{1}+r_{3}e_{2}+re_{3}+\dots+r_{k}e_{k})$
by (26). Now using the above argument again, we get
$f(s;re_{1}+re_{2}+r_{3}e_{3}+r_{4}e_{4}+\dots+r_{k}e_{k})=0$.

(b) We also use induction on $k$. Let first $d\geq 3$. For any $r\neq 0$,
choosing $y=s^{-1/n}re_{d}$ in (25), we have

$\displaystyle f\big{(}s^{1/n};s^{-1/n}re_{d}\big{)}=f\big{(}(\lambda
s)^{1/n};(\lambda
s)^{-1/n}re_{d}\big{)}+f\big{(}((1-\lambda)s)^{1/n};((1-\lambda)s)^{-1/n}re_{d}\big{)}$
(32)

for any $0<\lambda<1$ and $s>0$. Define

$\displaystyle\xi_{r}(s)=f(s^{1/n};s^{-1/n}re_{d})$

for $s>0$ and $r\in{\mathbb{R}}$. For arbitrary $a,b>0$, $r\in{\mathbb{R}}$,
set $s=a+b$, $\lambda=\frac{a}{a+b}$ in (32). We get that $\xi_{r}$ satisfies
the Cauchy functional equation. By the assumption, $\xi_{r}$ is regular. Hence

$\displaystyle f(s^{1/n};s^{-1/n}re_{d})=sf(1;re_{d})$

for any $s>0$ and $r\neq 0$. By (26), we finish the cases of $d\geq 3$ and
$k=1$.

For $k\geq 2$, we prove the case $d=2$ and the cases $d\geq 3$ together.
Assume that the desired results hold for $k-1$. If there are two different
non-zero coordinates of $x$, the induction assumption together with (26), (5),
(5) and (5) implies the desired result. If $r_{1}=r_{2}=\dots=r_{k}=:r$,
taking $y=s^{-1/n}(re_{1}+\dots+re_{k})$ in (25), we get

$\displaystyle f\left(s^{1/n};s^{-1/n}(re_{1}+\dots+re_{k})\right)$
$\displaystyle=$ $\displaystyle f\left((\lambda s)^{1/n};(\lambda
s)^{-1/n}(re_{1}+\dots+re_{k})\right)+f\left(((1-\lambda)s)^{1/n};((1-\lambda)s)^{-1/n}(re_{1}+\dots+re_{k})\right),$

which implies that the function $s\mapsto
f\left(s^{1/n};s^{-1/n}(re_{1}+\dots+re_{k})\right)$ satisfies the Cauchy
functional equation. Since it is regular,
$f\left(s^{1/n};s^{-1/n}(re_{1}+\dots+re_{k})\right)=sf\left(1;re_{1}+\dots+re_{k}\right)$.

(c) For $d\geq 3$, by (a), we only need to show that
$f\left(s^{1/n};s^{-1/n}(re_{1}+\dots+re_{m})\right)=0$ for any $1\leq m\leq
d$, which follows directly by the assumption in (c) and (28). The conclusion
for $d=2$ is similar. ∎

The following uniqueness theorems are also fundamental. The proofs are similar
to the corresponding proofs for $\mathrm{SL}(n)$ contravariant valuations
[28].

######  Lemma 11.

Let $Z$ and $Z^{\prime}$ be $\mathrm{SL}(n)$ covariant function-valued
valuations on $\mathcal{P}_{o}^{n}$. If $Z(sT^{d})=Z^{\prime}(sT^{d})$ for
every $s>0$ and $0\leq d\leq n$, then $ZP=Z^{\prime}P$ for every
$P\in\mathcal{P}_{o}^{n}$.

######  Lemma 12.

Let $Z$ and $Z^{\prime}$ be $\mathrm{SL}(n)$ covariant function-valued
valuations on $\mathcal{P}^{n}$. If $Z(sT^{d})=Z^{\prime}(sT^{d})$ and
$Z(s[e_{1},\dots,e_{d}])=Z^{\prime}(s[e_{1},\dots,e_{d}])$ for every $s>0$ and
$0\leq d\leq n$, then $ZP=Z^{\prime}P$ for every $P\in\mathcal{P}^{n}$.

##  6 Simple valuations

In this section, we first prove the following theorems.

######  Theorem 13.

Let $n\geq 3$. A map $Z:\mathcal{P}_{o}^{n}\to
C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$ is a regular, simple and
$\mathrm{SL}(n)$ covariant valuation if and only if there is a measure
$\mu\in\mathcal{M}^{c}({\mathbb{R}})$ such that

$\displaystyle ZP(x)=\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(P\cap
H_{x,t})d\mu(t)$

for every $P\in\mathcal{P}_{o}^{n}$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

Remark: for the “only if part”, we only need the weak assumption that
$Z:\mathcal{P}_{o}^{n}\to C({\mathbb{R}}^{n}\setminus\\{o\\})$ is a simple and
$\mathrm{SL}(n)$ covariant valuation such that $ZT^{n}\in
C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$ and the function $s\mapsto
Z(sT^{n})(s^{-1}x)$ is regular on $(0,\infty)$.

The characterization of $M_{\zeta}$ may be of special interest.

######  Theorem 14.

Let $n\geq 3$. A map $Z:\mathcal{P}_{o}^{n}\to
AC_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$ is a regular, simple and
$\mathrm{SL}(n)$ covariant valuation if and only if there is a locally
integrable function $\zeta$ on ${\mathbb{R}}$ such that

$\displaystyle ZP(x)=\int_{P}\zeta(x\cdot y)dy$

for every $P\in\mathcal{P}_{o}^{n}$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

For smooth $\zeta$, we obtain the following result.

######  Theorem 15.

Let $n\geq 3$. A map $Z:\mathcal{P}_{o}^{n}\to C^{\infty}({\mathbb{R}}^{n})$
is a regular, simple and $\mathrm{SL}(n)$ covariant valuation if and only if
there is a function $\zeta\in C^{\infty}({\mathbb{R}})$ such that

$\displaystyle ZP(x)$ $\displaystyle=\int_{P}\zeta(x\cdot y)dy$

for every $P\in\mathcal{P}_{o}^{n}$ and $x\in{\mathbb{R}}^{n}$.

The ”if” part of all theorems follows directly from Lemma 4.

Let

$\displaystyle
f(s;y)=Z(sT^{n})(y),~{}~{}y\in{\mathbb{R}}^{n}\setminus\\{o\\}.$ (33)

Since $Z$ is a simple and $\mathrm{SL}(n)$ covariant valuation, by (5), Lemma
10 (a) and Lemma 11, we only need the following lemma.

######  Lemma 16.

Let $n\geq 3$. If $Z:\mathcal{P}_{o}^{n}\to
C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$ is a regular, simple and
$\mathrm{SL}(n)$ covariant valuation, then there is a measure
$\mu\in\mathcal{M}^{c}({\mathbb{R}})$ such that

$\displaystyle
Z(sT^{n})(re_{1})=\frac{1}{|r|}\int_{{\mathbb{R}}}V_{n-1}(sT^{n}\cap
H_{re_{1},t})d\mu(t)$

for every $s>0$ and $r\neq 0$.

Moreover, if $Z(T^{n})\in AC_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$, there is
a locally integrable function $\zeta:{\mathbb{R}}\to{\mathbb{R}}$ such that

$\displaystyle Z(sT^{n})(re_{1})=\int_{sT^{n}}\zeta(re_{1}\cdot y)dy$

for every $s>0$ and $r\neq 0$. Further, $Z(T^{n})\in
C^{\infty}({\mathbb{R}}^{n})$ gives $\zeta\in C^{\infty}({\mathbb{R}})$ and
the above relation also holds for $r=0$.

###### Proof.

Define the function $g(r)=r^{n}Z(T^{n})(re_{1})$ on
${\mathbb{R}}\setminus\\{o\\}$ and let $g^{(k)}$ denote the $k$-th derivative
(if it exists) of $g$. We assume first that $g$ has the following property:
($\ast$) $g^{(k)}$ exists and is continuous on ${\mathbb{R}}\setminus\\{o\\}$
from $k=1$ up to $n-1$ and

$\displaystyle\lim_{r\to 0}g^{(k)}(r)=0$

for $0\leq k\leq n-1$.

By $(\ast)$, there is a signed Radon measure $\mu$ on ${\mathbb{R}}$ such that

$\displaystyle\int_{0}^{r}(r-t)^{n-1}d\mu(t)=(n-1)\int_{0}^{r}(r-t)^{n-2}g^{(n-1)}(t)dt.$
(34)

Indeed, let first $r>0$. We define the function
$j:{\mathbb{R}}\to{\mathbb{R}}$ by

$\displaystyle j(t)=\begin{cases}g^{(n-1)}(t),&t>0,\\\ 0,&t\leq 0.\end{cases}$

Property $(\ast)$ gives that $j$ is locally a BV function. By the structure
theorem of BV functions [12], there is a signed Radon measure $\mu$ such that

$\displaystyle\int_{{\mathbb{R}}}\theta(t)\mu(t)=-\int_{{\mathbb{R}}}j(t)\frac{\mathop{}\\!\mathrm{d}}{dt}\theta(t)dt$

for every $\theta\in C_{c}^{1}({\mathbb{R}})$, the set of compactly supported
and continuously differentiable functions on ${\mathbb{R}}$. Moreover, since
$j$ is continuous and $j(t)=0$ for all $t\leq 0$, the measure $\mu$ is
continuous and concentrated on $(0,\infty)$. Hence, we can choose $\theta\in
C_{c}^{1}({\mathbb{R}})$ such that
$\theta(t)=(r-t)^{n-1}\mathbbm{1}_{[o,r]}(t)$ for $t\geq 0$ and

$\displaystyle\int_{{\mathbb{R}}}\theta(t)\mu(t)$
$\displaystyle=\int_{0}^{r}(r-t)^{n-1}d\mu(t),$
$\displaystyle-\int_{{\mathbb{R}}}j(t)\frac{\mathop{}\\!\mathrm{d}}{dt}\theta(t)dt$
$\displaystyle=(n-1)\int_{0}^{r}(r-t)^{n-2}g^{(n-1)}(t)dt.$

Hence we get (34) for $r>0$. For $r<0$, we get similarly a signed and
continuous Radon measure $\nu$ concentrated on $(-\infty,0)$ satisfying (34).
Now taking the sum of $\mu$ and $\nu$, (34) still holds.

Using integration by parts, combined with $(\ast)$ and (34), we have

$\displaystyle\frac{1}{(n-1)!}\int_{0}^{r}(r-t)^{n-1}d\mu(t)$
$\displaystyle=\frac{1}{(n-2)!}\int_{0}^{r}(r-t)^{n-2}g^{(n-1)}(t)dt$
$\displaystyle=\frac{1}{(n-3)!}\int_{0}^{r}(r-t)^{n-3}g^{(n-2)}(t)dt$
$\displaystyle=\cdots=\int_{0}^{r}g^{(1)}(t)dt=g(r)=r^{n}Z(T^{n})(re_{1})$

for any $r\neq 0$. For any $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$, since the
function $s\mapsto Z(s^{1/n}T^{n})(s^{-1/n}x)$ is regular on $(0,\infty)$,
Lemma 10 (b) shows that

$\displaystyle Z(sT^{n})(s^{-1}x)=s^{n}Z(T^{n})(x)$ (35)

for any $s>0$. Hence

$\displaystyle Z(sT^{n})(re_{1})=s^{n}Z(T^{n})(rs)$
$\displaystyle=\frac{1}{(n-1)!r^{n}}\int_{0}^{rs}(rs-t)^{n-1}d\mu(t)$
$\displaystyle=\frac{1}{(n-1)!r}\int_{0}^{rs}(s-t/r)^{n-1}d\mu(t)$
$\displaystyle=\frac{1}{r}\int_{0}^{rs}V_{n-1}(sT^{n}\cap
H_{re_{1},t})d\mu(t)$
$\displaystyle=\frac{1}{|r|}\int_{-h_{sT}(-re_{1})}^{h_{sT}(re_{1})}V_{n-1}(sT^{n}\cap
H_{re_{1},t})d\mu(t)$
$\displaystyle=\frac{1}{|r|}\int_{{\mathbb{R}}}V_{n-1}(sT^{n}\cap
H_{re_{1},t})d\mu(t).$

Now we show that the function $g$ has the desired property $(\ast)$ if
$ZT^{n}\in C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$. We claim that the
function $r\mapsto rZT^{n}(re_{1}+\dots+re_{m})$ is differentiable on
${\mathbb{R}}\setminus\\{o\\}$ and

$\displaystyle\frac{\mathop{}\\!\mathrm{d}(rZT^{n}(re_{1}+\dots+re_{m}))}{\mathop{}\\!\mathrm{d}r}$
$\displaystyle=mZT^{n}(re_{1}+\dots+re_{m+1})-(m-1)ZT^{n}(re_{1}+\dots+re_{m})$
(36)

for any $r\neq 0$ and $1\leq m\leq n-1$. Indeed, set
$y_{k}=te_{3}+\dots+te_{k+2}+re_{k+3}+\dots+re_{m+1}$ for integer
$k\in[0,m-1]$. By (33), (35), (5) and (5),

$\displaystyle
ZT^{n}(re_{1}+te_{2}+y_{k})=\frac{rZT^{n}(re_{1}+y_{k})-tZT^{n}(te_{1}+y_{k})}{r-t}$
(37)

for any $r,t\neq 0$ such that $rt>0$. Take the sum of (37) over all $k$ from
$0$ to $m-1$ and let $t\to r$. By $ZT^{n}(\cdot)\in
C({\mathbb{R}}^{n}\setminus\\{o\\})$, we have

$\displaystyle mZT^{n}(re_{1}+\dots+re_{m+1})$ $\displaystyle=\lim_{t\to
r}\frac{rZT^{n}(re_{1}+y_{0})-tZT^{n}(te_{1}+y_{m-1})}{r-t}+\sum_{k=0}^{m-2}\frac{-tZT^{n}(te_{1}+y_{k})+rZT^{n}(re_{1}+y_{k+1})}{r-t}.$

Together with the permutation invariance of $ZT^{n}$, we get (6).

Let $\tilde{g}_{0}(r)=ZT^{n}(re_{1})$. The relation (6) shows that we can
define functions $\tilde{g}_{k}$ on ${\mathbb{R}}\setminus\\{o\\}$ for
$k=1,\dots n-1$ inductively by
$\tilde{g}_{k}(r)=\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}r}(r\tilde{g}_{k-1}(r))$
and $\tilde{g}_{k}(r)$ is a linear combination of
$ZT^{n}(re_{1}),\dots,ZT^{n}(re_{1}+\dots+re_{k+1})$. Using the chain rule
formula inductively, $g^{(k)}(r)$ is a linear combination of
$r^{n-k}\tilde{g}_{0}(r),\dots,r^{n-k}\tilde{g}_{k}(r)$ for $k=0,\dots n-1$.
Hence $g$ has the desired properties $(\ast)$ since $ZT^{n}\in
C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$.

For $ZT^{n}\in AC_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$, the argument in the
last paragraph shows that $g^{(n-1)}$ is absolutely continuous on any interval
in ${\mathbb{R}}$ after setting $g^{(n-1)}(0)=0$. Hence $\mu$ has a density
$\zeta$. The desired result follows from Fubini’s formula (2). For $ZT^{n}\in
C^{\infty}({\mathbb{R}}^{n})$, notice that now $g:r\mapsto
r^{n}Z(T^{n})(re_{1})\in C^{\infty}({\mathbb{R}})$. We just need to set
$\zeta(t)=g^{(n)}(t)$ for all $t\in{\mathbb{R}}$, and obtain $\zeta\in
C^{\infty}({\mathbb{R}})$. ∎

The following theorem follows directly from (33), (5), Lemma 10 (c) and Lemma
11.

######  Theorem 17.

Let $n\geq 3$. A map $Z:\mathcal{P}_{o}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ is a simple and $\mathrm{GL}_{+}(n)$
covariant valuation of weight $0$ if and only if

$\displaystyle ZP(x)=0$

for every $P\in\mathcal{P}_{o}^{n}$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

##  7 On lower-dimensional sets

In this section, we prove the following theorem.

######  Theorem 18.

Let $n\geq 3$. If $Z:\mathcal{P}_{o}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ is an $\mathrm{SL}(n)$ covariant
valuation and the function $r\mapsto ZT^{n-1}(\pm re_{n})$ is regular on
$(0,\infty)$, then there are functions $\zeta_{1},\zeta_{2}\in
F({\mathbb{R}})$ and a constant $c_{n-1}\in{\mathbb{R}}$ such that

$\displaystyle ZP(x)$
$\displaystyle=\zeta_{1}(h_{P}(x))+\zeta_{1}^{R}(h_{-P}(x))+E^{-}_{\zeta_{2}}(P)(x)+E^{-}_{\zeta_{2}^{R}}(-P)(x)+\frac{c_{n-1}}{(n-1)!|x|}V_{n-1}(P\cap
H_{x,o})$ (38)

for every $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$ and $P\in\mathcal{P}_{o}^{n}$
with $\dim P<n$.

Moreover, if $ZP\in C({\mathbb{R}}^{n}\setminus\\{o\\})$ for every
$P\in\mathcal{P}_{o}^{n}$, then $\zeta_{1},\zeta_{2}\in C({\mathbb{R}})$ and
$c_{n-1}=0$.

First, we show the following property of an $\mathrm{SL}(n)$ covariant map.

######  Lemma 19.

Let $n\geq 2$ and $Z:\mathcal{P}_{o}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ be an $\mathrm{SL}(n)$ covariant map. If
the linear hull of $P\in\mathcal{P}_{o}^{n}$ is
$\operatorname{lin}\\{e_{1},\dots,e_{d}\\}$ for some $0\leq d<n$, then

$\displaystyle
Z(sP)(x)=\begin{cases}ZP(s(r_{1}e_{1}+\dots+r_{d}e_{d})),&r_{1}e_{1}+\dots+r_{d}e_{d}\neq
o,\\\ ZP(e_{n}),&r_{1}e_{1}+\dots+r_{d}e_{d}=o,d\leq n-2,\\\
ZP(s^{-(n-1)}r_{n}e_{n}),&r_{1}e_{1}+\dots+r_{d}e_{d}=o,d=n-1\end{cases}$

for every $s>0$ and
$x=r_{1}e_{1}+\dots+r_{n}e_{n}\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

###### Proof.

First let $s=1$. Set $\phi:=\left[{\begin{array}[]{*{20}{c}}I_{d}&A\\\
0&B\end{array}}\right]\in\mathrm{SL}(n)$, where $I_{d}\in\mathbb{R}^{d\times
d}$ is the identity matrix, $A\in\mathbb{R}^{d\times(n-d)}$ is an arbitrary
matrix, $0\in\mathbb{R}^{(n-d)\times d}$ is the zero matrix,
$B\in\mathbb{R}^{(n-d)\times(n-d)}$ is an arbitrary matrix with the
determinant $1$. Also, let $x^{\prime}=r_{1}e_{1}+\dots+r_{d}e_{d}$ and
$x^{\prime\prime}=x-x^{\prime}$. Combining with $\phi P=P$ and the
$\mathrm{SL}(n)$ covariance of $Z$, we have

$\displaystyle Z(P)(x)=Z(\phi
P)(x)=Z(P)(\phi^{t}x)=Z(P)\left({\begin{array}[]{*{20}{c}}{x^{\prime}}\\\
{A^{t}x^{\prime}+B^{t}x^{\prime\prime}}\end{array}}\right).$

If $x^{\prime}\neq o$, we can choose appropriate matrices $A$ and $B$ such
that $A^{t}x^{\prime}+Bx^{\prime\prime}=0$. Hence $ZP(x)=ZP(x^{\prime})$ for
$x^{\prime}\neq 0$. If $x^{\prime}=o$, then $x^{\prime\prime}\neq o$. For
$d\leq n-2$, we can choose an appropriate matrix $B$ such that
$Bx^{\prime\prime}=e_{n}$ and then $Z(P)(x)=ZP(e_{n})$ for every
$x^{\prime}=o$. For $d=n-1$, the statement is trivial for $x^{\prime}=o$.
Hence we confirmed the case $s=1$ for arbitrary $x\neq o$.

Let $s>0$. The above argument gives that

$\displaystyle
ZP\left(sx^{\prime}+s^{-d/(n-d)}x^{\prime\prime}\right)=\begin{cases}ZP(sx^{\prime}),&x^{\prime}\neq
o,\\\ ZP(e_{n}),&x^{\prime}=o,d\leq n-2,\\\
ZP(s^{-(n-1)}r_{n}e_{n}),&x^{\prime}=o,d=n-1\end{cases}.$

Define $\psi\in\mathrm{SL}(n)$ by $\psi e_{i}=se_{i}$ for $1\leq i\leq d$ and
$\psi e_{j}=s^{-d/(n-d)}e_{j}$ for $d+1\leq j\leq n$. Since $Z$ is
$\mathrm{SL}(n)$ covariant, we have

$\displaystyle Z(sP)(x)=Z(\psi
P)(x)=ZP(\psi^{t}x)=ZP\left(sx^{\prime}+s^{-d/(n-d)}x^{\prime\prime}\right)$

Hence the desired statement holds for any arbitrary $s>0$. ∎

Now we consider $\mathrm{SL}(n)$ covariant valuations.

######  Lemma 20.

Let $Z:\mathcal{P}_{o}^{n}\to F({\mathbb{R}}^{n}\setminus\\{o\\})$ be an
$\mathrm{SL}(n)$ covariant valuation and let the function $r\mapsto
ZT^{n-1}(\pm re_{n})$ be regular on $(0,\infty)$. If $n\geq 3$, then there are
$c_{n-1},c\in{\mathbb{R}}$ such that

$\displaystyle ZT^{n-1}(re_{n})=c_{n-1}|r|^{-1}+c,~{}~{}ZT^{d}(re_{n})=c$

for any $r\neq 0$, $1\leq d\leq n-2$. If $n=2$ and $Z$ is $\mathrm{GL}(2)$
covariant of weight $0$, then there is $c\in{\mathbb{R}}$ such that

$\displaystyle ZT^{1}(re_{2})=c$

for any $r\neq 0$.

###### Proof.

First let $n=2$. For an arbitrary $r\neq 0$, we define $\phi\in\mathrm{GL}(2)$
by the following:

$\displaystyle\phi e_{2}=re_{2}\text{~{}and~{}}\phi e_{1}=e_{1}.$

Since $Z$ is $\mathrm{GL}(2)$ covariant of weight $0$, we have

$\displaystyle Z(T^{1})(e_{2})=Z(\phi T^{1})(e_{2})=Z(T^{1})(re_{2}),$

which implies the desired result.

Now let $n\geq 3$ and $r>0$. Combining Lemma 19 with (5) for $x=re_{n}$ and
$d\leq n-2$, we have
$Z(T^{d})(e_{n})=Z(T^{d})(re_{n})=Z(\widehat{T}^{d-1})(re_{n})$. Using the
$\mathrm{SL}(n)$ covariance of $Z$, we have

$\displaystyle Z(T^{n-2})(re_{n})=\dots=Z(T^{1})(re_{n})=:c.$

Now using (5) for $x=re_{n}$ and $d=n-1$ together with the $\mathrm{SL}(n)$
covariance of $Z$, we have

$\displaystyle
ZT^{n-1}(re_{n})+c=ZT^{n-1}(\lambda^{-1}re_{n})+ZT^{n-1}((1-\lambda)^{-1}re_{n})$
(39)

for any $r>0$. Set $g(r)=ZT^{n-1}(r^{-1}e_{n})$. For arbitrary $a,b>0$, we can
choose $r=(a+b)^{-1}$ and $\lambda=\frac{a}{a+b}$ in (39). Hence

$\displaystyle g(a+b)+c=g(a)+g(b)$

for any $a,b>0$. Now the regular function $g-c$ satisfying the Cauchy
functional equation implies that

$\displaystyle ZT^{n-1}(re_{n})=c_{n-1}r^{-1}+c$

for any $r>0$, where $c_{n-1}$ is a constant.

Since $Z$ is $\mathrm{SL}(n)$ covariant, we have
$ZT^{d}(re_{n})=ZT^{d}(-re_{n})$ for every $d\leq n-1$ if $n\geq 3$, which
confirms the case $r<0$. ∎

######  Lemma 21.

For $n\geq 3$, let $Z:\mathcal{P}_{o}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ be an $\mathrm{SL}(n)$ covariant
valuation and let the function $r\mapsto ZT^{n-1}(\pm re_{n})$ be regular on
$(0,\infty)$. If

$\displaystyle
Z\\{o\\}(e_{n})=ZT^{1}(re_{1})=ZT^{2}(re_{1})=ZT^{n-1}(re_{n})=0$ (40)

for every $r\neq 0$, then $Z$ is simple.

###### Proof.

By Lemma 11 and Lemma 19, we only need to show that $Z(T^{d})(x)=0$ for any
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$ and $0\leq d\leq n-1$. Lemma 19
together with $Z\\{o\\}(e_{n})=0$ implies that it holds for $d=0$.

Lemma 20 implies that $ZT^{n-1}(re_{n})=c_{n-1}|r|^{-1}+c$ and
$ZT^{k}(re_{n})=c$ for any $r\neq 0$ and $1\leq k\leq n-2$. Combined it with
the assumption that $ZT^{n-1}(re_{n})=0$ for any $r\in{\mathbb{R}}$, we have

$\displaystyle ZT^{d}(re_{n})=0$ (41)

for all $1\leq d\leq n-1$ and $r\neq 0$. Hence, together with (40), Lemma 19
implies that our statement holds for $d=1$.

Now let $2\leq d\leq n-1$. Assume that our statement holds for $d-1$.

Set

$\displaystyle
f(s;r_{1}e_{1}+\dots+r_{d}e_{d})=Z(sT^{d})(r_{1}e_{1}+\dots+r_{d}e_{d}).$ (42)

Together with the induction assumption, (5) and the $\mathrm{SL}(n)$
covariance of $Z$, we get the equation (25). Also, Lemma 19 implies that
$f(s^{1/n};s^{-1/n}x)=f(x)$ for any $s>0$,
$x\in{\mathbb{R}}^{d}\setminus\\{o\\}$ and $0\leq d\leq n-1$. Now Lemma 10
(c), together with the assumption $ZT^{2}(re_{1})=0$ (only needed for $d=2$)
gives that $ZT^{d}(x)=0$ for $x\in{\mathbb{R}}^{d}\setminus\\{o\\}$.
Generally, for $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$, we write
$x=x^{\prime}+x^{\prime\prime}$ as in the proof of Lemma 19. By (41), we have
$Z(T^{d})(re_{n})=0$ for all $r\in{\mathbb{R}}\setminus\\{o\\}$. Together with
Lemma 19, we get $Z(sT^{d})(x)=0$ for $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.
∎

###### Proof of Theorem 18.

By Lemma 20, there is a $c_{n-1}$ such that

$\displaystyle ZT^{n-1}(re_{n})=c_{n-1}|r|^{-1}+ZT^{1}(e_{n}).$

for any $r\neq 0$. Define $\zeta_{1},\zeta_{2}$ by

$\displaystyle\zeta_{1}(r)=\begin{cases}ZT^{2}(re_{1})-\frac{1}{2}ZT^{1}(e_{n}),&r\neq
0,\\\ \frac{1}{2}ZT^{1}(e_{n}),&r=0,\end{cases}$ (43)
$\displaystyle\zeta_{2}(r)=\begin{cases}ZT^{2}(re_{1})-ZT^{1}(re_{1})+\frac{1}{2}(Z\\{o\\}(e_{n})-ZT^{1}(e_{n})),&r\neq
0,\\\ \frac{1}{2}(Z\\{o\\}(e_{n})-ZT^{1}(e_{n})),&r=0.\end{cases}$ (44)

Now we consider a new valuation which is the difference of $Z$ and the
valuation given in Theorem 18 (with $\zeta_{1},\zeta_{2}$ and $c_{n-1}$
defined above). Elementary calculations show that the new valuation satisfies
(40). Hence the statement of the first part follows from Lemma 21.

If $ZT^{1},ZT^{2}\in C({\mathbb{R}}^{n}\setminus\\{o\\})$, the definitions of
$\zeta_{1},\zeta_{2}$ show that $\zeta_{1},\zeta_{2}\in
C({\mathbb{R}}\setminus\\{o\\})$. Let $-1<r<1$. The statement of the first
part shows that

$\displaystyle Z[o,e_{2}](-e_{1}+re_{2})$
$\displaystyle=\zeta_{1}(r)+\zeta_{1}(0)-\zeta_{2}(r)+\zeta_{2}(0),$
$\displaystyle Z[0,e_{1},e_{2}](-e_{1}+re_{2})$
$\displaystyle=\begin{cases}\zeta_{1}(r)+\zeta_{1}(-1),&0\leq r<1,\\\
\zeta_{1}(0)+\zeta_{1}(-1)-\zeta_{2}(r)+\zeta_{2}(0),&-1<r<0.\end{cases}$

Since $Z[o,e_{2}](-e_{1}+re_{2})\to Z[o,e_{2}](-e_{1})$ and
$Z[0,e_{1},e_{2}](-e_{1}+re_{2})\to Z[0,e_{1},e_{2}](-e_{1})$ as $r\to 0$, we
find that $\zeta_{1},\zeta_{2}$ are continuous at $0$. Hence
$\zeta_{1},\zeta_{2}\in C({\mathbb{R}})$.

Now, only the function $x\mapsto\frac{1}{|x|}V_{n-1}(T^{n-1}\cap H_{x,o})$ is
not continuous on ${\mathbb{R}}^{n}\setminus\\{o\\}$. Thus $c_{n-1}=0$. ∎

##  8 Proofs of Theorems 1-3

The “if” parts of Theorems 1-3 are already completed by §4. Hence we only deal
with the “only if” parts.

###### Proof of Theorem 2.

Since $ZT^{n-1}\in C({\mathbb{R}}^{n}\setminus\\{o\\})$, the function
$r\mapsto ZT^{n-1}(\pm re_{n})$ is regular on $(0,\infty)$. Hence Theorem 18
gives

$\displaystyle ZP(x)$
$\displaystyle=\zeta_{1}(h_{P}(x))+\zeta_{1}^{R}(h_{-P}(x))+E^{-}_{\zeta_{2}}(P)(x)+E^{-}_{\zeta_{2}^{R}}(-P)(x)$
(45)

for every $x\in{\mathbb{R}}^{n}$ and $P\in\mathcal{P}_{o}^{n}$ with $\dim
P<n$, where $\zeta_{1},\zeta_{2}\in C({\mathbb{R}})$. Let
$\tilde{Z}P(x)=ZP(x)-(\zeta_{1}(h_{P}(x))+\zeta_{1}^{R}(h_{-P}(x))+E^{-}_{\zeta_{2}}(P)(x)+E^{-}_{\zeta_{2}^{R}}(-P)(x))$.
Clearly $\tilde{Z}:\mathcal{P}_{o}^{n}\to
C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$ is a regular, simple and
$\mathrm{SL}(n)$ covariant valuation. Now the desired result follows from
Theorem 13. ∎

Before proving Theorem 3, we first show the following.

######  Lemma 22.

Let $n=2$ and $Z:\mathcal{P}_{o}^{2}\to F({\mathbb{R}}^{2}\setminus\\{o\\})$
be a $\mathrm{GL}(2)$ covariant valuation of weight $0$. If

$\displaystyle Z\\{o\\}(e_{2})=ZT^{1}(re_{1})=ZT^{2}(re_{1})=ZT^{1}(e_{2})=0$

for every $r\neq 0$, then $Z=0$.

###### Proof.

Since we assume that $Z$ is $\mathrm{GL}(2)$ covariant of weight $0$, together
with Lemma 20 (for $n=2$), the proof is the same as the proof of Lemma 21 (for
$n=3$). ∎

###### Proof of Theorem 3.

Let first $n\geq 3$. Since $Z$ is $\mathrm{GL}_{+}(n)$ covariant of weight
$0$, $r\mapsto ZT^{n-1}(\pm re_{n})=ZT^{n-1}(e_{1})$ is regular on
$(0,\infty)$. Hence by Theorem 18, (38) holds. Only the valuation
$P\mapsto\frac{1}{|x|}V_{n-1}(P\cap H_{x,o})$ is not $\mathrm{GL}_{+}(n)$
covariant of weight $0$. Similar to the proof of Theorem 2, the desired result
follows from Theorem 17.

Let $n=2$. By Lemma 22, the argument is similar to the proof of Theorem 18. ∎

The following lemma will be used in the proof of Theorem 1.

######  Lemma 23.

Let $\zeta_{1},\zeta_{2}\in C({\mathbb{R}})$ and $Z:\mathcal{P}_{o}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ defined by (45) for every
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$ and $P\in\mathcal{P}_{o}^{n}$ with
$\dim P<n$. If $Z$ is continuous on $\mathcal{P}_{o}^{n}$, then

$\displaystyle\zeta_{2}\equiv 0.$

###### Proof.

Let $r\geq 0$, $s,t>0$ and $P_{r}=[-te_{1},e_{1}]+[-e_{2},re_{2}]$. By (45),
we have

$\displaystyle ZP_{r}(se_{1})$
$\displaystyle=\zeta_{1}(s)+\zeta_{1}(-st)+\zeta_{2}(s)+\zeta_{2}(-st)$

for $r>0$ and

$\displaystyle ZP_{0}(se_{1})$ $\displaystyle=\zeta_{1}(s)+\zeta_{1}(-st)$

for $t>0$. Since $ZP_{r}(se_{1})\mapsto ZP_{0}(se_{1})$ for any $s>0$, we have

$\displaystyle\zeta_{2}(s)+\zeta_{2}(-st)=0$ (46)

for any $s,t>0$. Since $\zeta_{2}\in C({\mathbb{R}})$, taking $s\mapsto 0$ in
(46), we have $\zeta_{2}(0)=0$. Next, taking $t\mapsto 0$, we have
$\zeta_{2}(s)=0$ for any $s>0$. Back to (46), we get $\zeta_{2}(s)=0$ for any
$s<0$. ∎

###### Proof of Theorem 1.

Lemma 9 shows that $ZP\in C({\mathbb{R}}^{n}\setminus\\{o\\})$ for every
$P\in\mathcal{P}_{o}^{n}$. Hence the same argument as in the proof of Theorem
2 shows that (45) holds for $\zeta_{1},\zeta_{2}\in C({\mathbb{R}})$. Together
with the continuity of $Z$, by Lemma 23, we have $\zeta_{2}\equiv 0$.

Now it follows from Lemma 5 and Lemma 6 that the map $\bar{Z}:P\mapsto
ZP(x)-(\zeta_{1}(h_{P}(x))+\zeta_{1}^{R}(h_{-P}(x))$ is a continuous, simple
and $\mathrm{SL}(n)$ covariant valuation. By the remark after Theorem 13, we
only need to show that $s\mapsto\bar{Z}(sT^{n})(s^{-1}x)$ is regular on
$(0,\infty)$ and $\lim_{s\to 0}s\bar{Z}(T^{n})(sx)=0$ for any
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$. The last follows from Lemma 10 (b)
(that (35) holds) and Lemma 9. ∎

##  9 Euler-type relations

The maps $E^{\pm}_{\zeta},E_{\zeta}$ are all $\mathrm{GL}(n)$ covariant
valuations of weight $0$; see Lemma 8. This fact together with Theorem 3 gives
a new proof of the following Euler-type relation (for all $\mathcal{P}^{n}$).

######  Corollary 24.

For $n\geq 0$, the Euler-type relation

$\displaystyle\sum_{F\in\mathcal{F}(P)}(-1)^{\dim
F}\zeta\left(h_{F}(x)\right)=\zeta(-h_{-P}(x))$ (47)

holds for every $P\in\mathcal{P}^{n}$, $\zeta\in F({\mathbb{R}})$ and
$x\in{\mathbb{R}}^{n}$.

Shephard [48] established this Euler-type relation for linear functions
$\zeta$. His proof also works for arbitrary $\zeta$. For linear $\zeta$,
McMullen [41] established Euler-type relations for an arbitrary function-
valued valuation $Z$ with the additional assumption that
$Z(P+y)(x)=ZP(x)+v(P,x)\cdot y$ for all $y\in{\mathbb{R}}^{n}$ where $v(P,x)$
is any vector.

Notice that (47) for $\zeta\equiv 1$ is the Euler-Schläfli-Poincaré formula

$\displaystyle\sum_{F\in\mathcal{F}(P)}(-1)^{\dim F}V_{0}(F)=\sum_{j=0}^{\dim
P}(-1)^{j}|\mathcal{F}_{j}(P)|=V_{0}(P),$

where $V_{0}$ is the Euler characteristic, which is $1$ for any convex body
and $0$ for the empty set.

###### Proof of Corollary 24.

The definition of $E_{\zeta}=\sum_{F\in\mathcal{F}(P)}(-1)^{\dim
F}\zeta\left(h_{F}\right)$ does not rely on the dimension of the ambient
space, so we can set the ambient space as ${\mathbb{R}}^{n}$ for $n\geq 2$.

For $x=o$, (47) is either the trivial case $0=0$ or the Euler-Schläfli-
Poincaré formula. The last can be also obtained by setting $\zeta\equiv 1$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$ in (47). Hence we only need to prove
(47) for $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

Lemma 8 shows that
$E_{\zeta}:\mathcal{P}_{o}^{n}\to\mathcal{F}({\mathbb{R}}^{n})$ is a
$\mathrm{GL}(n)$ covariant valuation of weight $0$. Hence $E_{\zeta}$ is
described in Theorem 3, that is,

$\displaystyle
E_{\zeta}P(x)=\zeta_{1}(h_{P}(x))+\zeta_{1}^{R}(h_{-P}(x))+E^{-}_{\zeta_{2}}(P)(x)+E^{-}_{\zeta_{2}^{R}}(-P)(x).$
(48)

with some $\zeta_{1},\zeta_{2}\in F({\mathbb{R}})$ for all
$P\in\mathcal{P}_{o}^{n}$. By the definition of $E_{\zeta}$, we get that

$\displaystyle
E_{\zeta}T^{2}(re_{1})=E_{\zeta}T^{1}(re_{1})=\begin{cases}\zeta(0),&r>0,\\\
\zeta(r),&r<0,\end{cases}$

and

$\displaystyle E_{\zeta}T^{1}(e_{n})=E_{\zeta}\\{o\\}(e_{n})=\zeta(0).$

Since $\zeta_{1},\zeta_{2}$ are determined by (43), (44) for $Z=E_{\zeta}$, we
have

$\displaystyle\zeta_{1}(r)=\begin{cases}\frac{1}{2}\zeta(0),&r\geq 0,\\\
\zeta(r)-\frac{1}{2}\zeta(0),&r<0,\end{cases}$

and $\zeta_{2}\equiv 0$. For $o\in P$, we have $h_{P}(x),h_{-P}(x)\geq 0$.
Hence (48) implies that (47) holds for $o\in P$.

Let $x_{1},\dots,x_{m}$ be the vertices of $P\in\mathcal{P}_{o}^{n}$ and let
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$. By the definition of support
functions, we have $h_{F}(x)=x\cdot x_{j(F)}$ for all $F\in\mathcal{F}(P)$ and
$-h_{-P}(x)=x\cdot x_{\tilde{j}(P)}$ with some $1\leq j(F),\tilde{j}(P)\leq
m$. Hence we can reformulate (47) for these $P$ and $x$ as

$\displaystyle 0=\zeta(-h_{-P}(x))-\sum_{F\in\mathcal{F}(P)}(-1)^{\dim
F}\zeta\left(h_{F}(x)\right)=\sum_{i=1}^{m}a_{i}\zeta(x\cdot x_{i}),$ (49)

where

$\displaystyle
a_{i}=\mathbbm{1}_{i}(\tilde{j}(P))-\sum_{F\in\mathcal{F}(P)}(-1)^{\dim
F}\mathbbm{1}_{i}(j(F)).$ (50)

Now we claim that $a_{i}=0$ for every $1\leq i\leq m$. Indeed, assume first
that $x\cdot x_{i}$ are all different. Since ${\mathbb{R}}$ is an infinite
dimensional space over the rational field ${\mathbb{Q}}$, we can find
$r_{1},\dots,r_{m}\in{\mathbb{R}}\setminus\\{o\\}$ such that they are
${\mathbb{Q}}$-independent. Also, $a_{i}$ are integers and do not rely on the
choice of $\zeta$, so we choose $\zeta\in F({\mathbb{R}})$ such that
$\zeta(x\cdot x_{i})=r_{i}$. Thus (49) gives that all $a_{i}=0$. If some of
$x\cdot x_{i}$ are the same, we just need that the (different) elements of the
set $\\{\zeta(x\cdot x_{i})\\}$ to be nonzero ${\mathbb{Q}}$-independent
numbers.

The last part is to show that (47) holds for $P+y$ for all $y\in{\mathbb{R}}$.
The vertices of $P+y$ are $x_{1}+y,\dots,x_{m}+y$. Notice that
$h_{F+y}(x)=x\cdot(x_{j(F)}+y)$ and
$-h_{-(P+y)}(x)=x\cdot(x_{\tilde{j}(P)}+y)$. We have

$\displaystyle\zeta(-h_{-(P+y)}(x))-\sum_{F\in\mathcal{F}(P+y)}(-1)^{\dim
F}\zeta\left(h_{F}(x)\right)=\sum_{i=1}^{m}a_{i}\zeta(x\cdot(x_{i}+y)).$

Hence (47) holds for all $P+y$. Since $P$ and $x$ are chosen arbitrary in the
above argument, the proof is completed. ∎

Corollary 24 immediately implies the following.

######  Corollary 25.

For $n\geq 1$, we have

$\displaystyle\sum_{F\in\mathcal{F}(P)}(-1)^{\dim
F}\int_{S^{n-1}}\zeta\left(\frac{h_{F}(u)}{h_{L}(u)}\right)dV_{L}(u)=\int_{S^{n-1}}\zeta\left(-\frac{h_{-P}(u)}{h_{L}(u)}\right)dV_{L}(u)$

for every $\zeta\in F({\mathbb{R}})$, $P\in\mathcal{P}^{n}$ and
$L\in\mathcal{K}^{n}$ such that $o$ is contained in the interior of $L$.

Here $V_{L}$ is the cone volume measure of $L$, that is,
$V_{L}(\omega)=\frac{1}{n}\int_{\nu_{K}^{-1}(\omega)}h_{K}(\nu_{K}(x))d\mathcal{H}^{n-1}(x)$
for any Borel set $\omega\subset S^{n-1}$. Here $\nu_{K}$ is the Gauss map of
the boundary of $K$ and $\mathcal{H}^{n-1}$ is the $(n-1)$-dimensional
Hausdorff measure.

We also obtain local versions of the Euler-Schläfli-Poincaré formula related
to $E^{\pm}_{\zeta}(P)$. Let $1$ denote the constant function $1$ on
${\mathbb{R}}$. By the definition, $E^{\pm}_{1}(P)=\sum_{j=0}^{\dim
P}(-1)^{j}|\mathcal{F}^{\pm}_{j}(P)|$. If further $o\in P$, then
$E^{-}_{1}(P)=\sum_{F\in\mathcal{F}(P)}(-1)^{\dim F}V_{0}(o\cap F)$. We obtain
the following result.

######  Corollary 26 (The local Euler-Schläfli-Poincaré formula).

For $n\geq 0$, we have

$\displaystyle\sum_{j=0}^{\dim P}(-1)^{j}|\mathcal{F}^{-}_{j}(P)|=(-1)^{\dim
P}V_{0}(o\cap\operatorname{relint}P),$ $\displaystyle\sum_{j=0}^{\dim
P}(-1)^{j}|\mathcal{F}^{+}_{j}(P)|=V_{0}(o\cap P),$

and

$\displaystyle\sum_{F\in\mathcal{F}(P)}(-1)^{\dim F}V_{0}(x\cap F)=(-1)^{\dim
P}V_{0}(x\cap\operatorname{relint}P)$

for every $x\in{\mathbb{R}}^{n}$ and $P\in\mathcal{P}^{n}$.

The last formula was also established by Kabluchko, Last, and Zaporozhets
[24]. They even established a general formula, replacing $x$ by an arbitrary
affine subspace (the right side needs to be changed a little bit in the
general case). Using such general formula, they confirmed the so-called Cowan
formula (identity).

To prove Corollary 26, we need introduce the following lemma.

######  Lemma 27 (Ludwig and Reitzner [36]).

Let $n\geq 2$. A map $Z:\mathcal{P}^{n}\to{\mathbb{R}}$ is a $\mathrm{GL}(n)$
invariant valuation of weight $0$ if and only if there are constants
$c_{1},c_{2},c_{3}\in{\mathbb{R}}$ such that

$\displaystyle ZP=c_{1}V_{0}(P)+c_{2}(-1)^{\dim
P}V_{0}(o\cap\operatorname{relint}P)+c_{3}V_{0}(o\cap P)$

for every $P\in\mathcal{P}^{n}$.

###### Proof of Corollary 26.

Since desired formulae do not rely on the choice of the dimension of the
ambient space, we can set the ambient space as ${\mathbb{R}}^{n}$ for $n\geq
2$. By Lemma 8, the left sides of the first two desired formulae are
$\mathrm{GL}(n)$ invariant valuations of weight $0$. Together with Lemma 27,
we have

$\displaystyle\sum_{j=0}^{\dim
P}(-1)^{j}|\mathcal{F}^{\pm}_{j}(P)|=c_{1}^{\pm}V_{0}(P)+c_{2}^{\pm}(-1)^{\dim
P}V_{0}(o\cap\operatorname{relint}P)+c_{3}^{\pm}V_{0}(o\cap P).$

with $c_{1}^{\pm},c_{2}^{\pm},c_{3}^{\pm}$. Choosing $P$ as $\\{o\\}$,
$[-e_{1},e_{1}]$ or $[e_{1},2e_{1}]$, we get

$\displaystyle\begin{cases}c_{1}^{-}+c_{2}^{-}+c_{3}^{-}=1,\\\
c_{1}^{-}-c_{2}^{-}+c_{3}^{-}=-1,\\\
c_{1}^{-}=0,\end{cases}~{}~{}~{}~{}\begin{cases}c_{1}^{+}+c_{2}^{+}+c_{3}^{+}=1,\\\
c_{1}^{+}-c_{2}^{+}+c_{3}^{+}=1,\\\ c_{1}^{+}=0.\end{cases}$

Thus,

$\displaystyle c_{1}^{-}=0,~{}c_{2}^{-}=1,~{}c_{3}^{-}=0,$ $\displaystyle
c_{1}^{+}=0,~{}c_{2}^{+}=0,~{}c_{3}^{+}=1,$

Hence the first two desired formulae are true. For $o\in P$, the first formula
is equivalent to the following

$\displaystyle\sum_{F\in\mathcal{F}(P)}(-1)^{\dim F}V_{0}(o\cap F)=(-1)^{\dim
P}V_{0}(o\cap\operatorname{relint}P).$ (51)

Clearly, (51) also holds for $o\notin P$. Now replacing $P$ by $P-x$ in (51),
we get the third formula. ∎

##  10 Valuations on $\mathcal{P}^{n}$ and $\mathcal{K}^{n}$

To extend Theorem 2 to $\mathcal{P}^{n}$, the natural idea is to check whether
the representation

$\displaystyle\zeta_{1}(h_{P}(x))+\zeta_{1}^{R}(h_{-P}(x))+E^{-}_{\zeta_{2}}(P)(x)+E^{-}_{\zeta_{2}^{R}}(-P)(x)+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(P\cap
H_{x,t})d\mu(t)$
$\displaystyle\qquad+\tilde{\zeta}_{1}(h_{[P,o]}(x))+\tilde{\zeta}_{1}^{R}(h_{-[P,o]}(x))+E^{-}_{\tilde{\zeta}_{2}}([P,o])(x)+E^{-}_{\tilde{\zeta}_{2}^{R}}(-[P,o])(x)$
$\displaystyle\qquad+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}([P,o]\cap
H_{x,t})d\tilde{\mu}(t)$

gives all $\mathrm{SL}(n)$ covariant
$C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$-valued valuations, where
$\zeta_{1},\zeta_{2},\tilde{\zeta}_{1},\tilde{\zeta}_{2}\in C({\mathbb{R}})$
and $\mu,\tilde{\mu}\in\mathcal{M}^{c}({\mathbb{R}})$. Unfortunately, it does
not. Instead, replacing $\zeta^{R}(h_{-P})$ by $E^{+}_{\zeta}(P)$ (by
Corollary 24) in Theorem 2 gives the idea of the following result.

######  Theorem 28.

Let $n\geq 3$. A map $Z:\mathcal{P}^{n}\to
C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$ is a regular and $\mathrm{SL}(n)$
covariant valuation if and only if there are
$\zeta_{1},\zeta_{2},\tilde{\zeta}_{1},\tilde{\zeta}_{2}\in C({\mathbb{R}})$
and $\mu,\tilde{\mu}\in\mathcal{M}^{c}({\mathbb{R}})$ such that

$\displaystyle ZP(x)$
$\displaystyle=E^{+}_{\zeta_{1}}(P)(x)+E^{+}_{\zeta_{1}^{R}}(-P)(x)+E^{-}_{\zeta_{2}}(P)(x)+E^{-}_{\zeta_{2}^{R}}(-P)(x)+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(P\cap
H_{x,t})d\mu(t)$
$\displaystyle\qquad+E^{+}_{\tilde{\zeta}_{1}}([P,o])(x)+E^{+}_{\tilde{\zeta}_{1}^{R}}(-[P,o])(x)+E^{-}_{\tilde{\zeta}_{2}}([P,o])(x)+E^{-}_{\tilde{\zeta}_{2}^{R}}(-[P,o])(x)$
$\displaystyle\qquad+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}([P,o]\cap
H_{x,t})d\tilde{\mu}(t)$ (52)

for every $P\in\mathcal{P}^{n}$ and $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

###### Proof.

Let $\mathcal{T}_{o}^{n}$ be the set of simplices in ${\mathbb{R}}^{n}$ with
one vertex at the origin. In the proof of Theorem 2, we have shown that a
regular and $\mathrm{SL}(n)$ covariant valuation $Z:\mathcal{T}_{o}^{n}\to
C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$ can be described as

$\displaystyle Z(sT^{d})(x)$
$\displaystyle=\eta_{1}(h_{P}(x))+\eta_{1}^{R}(h_{-P}(x))+E^{-}_{\eta_{2}}(sT^{d})(x)+E^{-}_{\eta_{2}^{R}}(-sT^{d})(x)$
$\displaystyle\qquad+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(sT^{d}\cap
H_{x,t})d\nu(t)$

for any $s>0$, $0\leq d\leq n$ and $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$ with
some $\eta_{1},\eta_{2}\in C({\mathbb{R}})$ and
$\nu\in\mathcal{M}^{c}({\mathbb{R}})$. Further by Corollary 24 and
$E^{+}_{\eta}P(x)=\sum_{F\in\mathcal{F}(P)}(-1)^{\dim
F}\eta\left(h_{F}(x)\right)$ when $o\in P$, we get

$\displaystyle Z(sT^{d})(x)$
$\displaystyle=E^{+}_{\eta_{1}}(sT^{d})(x)+E^{+}_{\eta_{1}^{R}}(-sT^{d})(x)+E^{-}_{\eta_{2}}(sT^{d})(x)+E^{-}_{\eta_{2}^{R}}(-sT^{d})(x)$
$\displaystyle\qquad+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(sT^{d}\cap
H_{x,t})d\nu(t).$ (53)

For any $T\in\mathcal{T}_{o}^{n}\setminus\\{o\\}$, we write $\tilde{T}$ for
its $(\dim T-1)$-dimensional face opposite to the origin. We define the new
map $\tilde{Z}:\mathcal{T}_{o}^{n}\to C_{r}({\mathbb{R}}^{n}\setminus\\{o\\})$
by $\tilde{Z}T=Z\tilde{T}$ for any $T\in\mathcal{T}_{o}^{n}\setminus\\{o\\}$
and $\tilde{Z}\\{o\\}=0$. It is not hard to show that $\tilde{Z}$ is also a
regular and $\mathrm{SL}(n)$ covariant valuation on $\mathcal{T}_{o}^{n}$.
Hence, there are $\tilde{\eta}_{1},\tilde{\eta}_{2}\in C({\mathbb{R}})$ and
$\tilde{\nu}\in\mathcal{M}^{c}({\mathbb{R}})$ such that

$\displaystyle Z(s[e_{1},\dots,e_{d}])(x)$ $\displaystyle=\tilde{Z}(sT^{d})$
$\displaystyle=E^{+}_{\tilde{\eta}_{1}}(sT^{d})(x)+E^{+}_{(\tilde{\eta}_{1})^{R}}(-sT^{d})(x)+E^{-}_{\tilde{\eta}_{2}}(sT^{d})(x)+E^{-}_{(\tilde{\eta}_{2})^{R}}(-sT^{d})(x)$
$\displaystyle\qquad+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(sT^{d}\cap
H_{x,t})d\tilde{\nu}(t)$ (54)

for any $s>0$, $1\leq d\leq n$ and $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$. Set
$\tilde{\zeta}_{i}=\tilde{\eta}_{i}$ and $\zeta_{i}=\eta_{i}-\tilde{\eta}_{i}$
for $i=1,2$ and $\tilde{\mu}=\tilde{\nu},~{}\mu=\nu-\tilde{\nu}$. The
valuation defined by the right side of (28) satisfies (10) and (10). Now the
proof is completed by Lemma 12. ∎

The extension of Theorem 3 to $\mathcal{P}^{n}$ is similar to Theorem 28.

######  Theorem 29.

Let $n\geq 3$. A map $Z:\mathcal{P}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ is a $\mathrm{GL}_{+}(n)$ covariant
valuation if and only if there are
$\zeta_{1},\zeta_{2},\tilde{\zeta}_{1},\tilde{\zeta}_{2}\in F({\mathbb{R}})$
such that

$\displaystyle ZP(x)$
$\displaystyle=E^{+}_{\zeta_{1}}(P)(x)+E^{+}_{\zeta_{1}^{R}}(-P)(x)+E^{-}_{\zeta_{2}}(P)(x)+E^{-}_{\zeta_{2}^{R}}(-P)(x)$
$\displaystyle\qquad+E^{+}_{\tilde{\zeta}_{1}}([P,o])(x)+E^{+}_{\tilde{\zeta}_{1}^{R}}(-[P,o])(x)+E^{-}_{\tilde{\zeta}_{2}}([P,o])(x)+E^{-}_{\tilde{\zeta}_{2}^{R}}(-[P,o])(x)$

for every $P\in\mathcal{P}^{n}$ and $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

A map $Z:\mathcal{P}^{2}\to F({\mathbb{R}}^{2}\setminus\\{o\\})$ is a
$\mathrm{GL}(2)$ covariant valuation of weight $0$ if and only if the above
relation holds for $n=2$.

Some readers may wonder why the valuation $\zeta(h_{P})$ seems to be missing
in Theorem 28. In fact, it is included in the function classified in Theorem
28 as

$\displaystyle
E^{-}_{\zeta}P-E^{-}_{\zeta}[P,o]=\zeta(-h_{-P})-\zeta(-h_{[-P,o]})$ (55)

This formula will be proved in the Theorem 31.

Analogs to Lemma 23, the following lemma will be used in the proof of Theorem
31.

######  Lemma 30.

Let $\zeta_{1},\tilde{\zeta}_{1},\zeta_{2},\tilde{\zeta}_{2}\in
C({\mathbb{R}})$ and $Z:\mathcal{P}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ defined by

$\displaystyle ZP(x)$
$\displaystyle=E^{+}_{\zeta_{1}}(P)(x)+E^{+}_{\zeta_{1}^{R}}(-P)(x)+E^{-}_{\zeta_{2}}(P)(x)+E^{-}_{\zeta_{2}^{R}}(-P)(x)$
$\displaystyle\qquad+E^{+}_{\tilde{\zeta}_{1}}([P,o])(x)+E^{+}_{\tilde{\zeta}_{1}^{R}}(-[P,o])(x)+E^{-}_{\tilde{\zeta}_{2}}([P,o])(x)+E^{-}_{\tilde{\zeta}_{2}^{R}}(-[P,o])(x)$
(56)

for every $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$, $P\in\mathcal{P}_{o}^{n}$
with $\dim P<n$ and for every $P\in\mathcal{P}^{n}$ with $\dim P<n-1$. If $Z$
is continuous, then

$\displaystyle\zeta_{2}+\tilde{\zeta}_{2}\equiv 0,~{}\zeta_{1}\equiv 0.$

###### Proof.

By (7), Corollary 24, the description (30) shows that

$\displaystyle ZP(x)$
$\displaystyle=(\zeta_{1}+\tilde{\zeta}_{1})(h_{P}(x))+(\zeta_{2}+\tilde{\zeta}_{2})(h_{-P}(x))+E^{-}_{\zeta_{2}+\tilde{\zeta}_{2}}(P)(x)+E^{-}_{(\zeta_{2}+\tilde{\zeta}_{2})^{R}}(-P)(x)$

for every $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$ and $P\in\mathcal{P}_{o}^{n}$
with $\dim P<n$. Hence the relation $\zeta_{2}+\tilde{\zeta}_{2}\equiv 0$
follows from Lemma 23.

Let $r\geq 0$, $s,t>0$ and $P_{r}=[-te_{1},e_{1}]+re_{2}$. The description
(30) gives that

$\displaystyle ZP_{r}(se_{1})=\tilde{\zeta}_{1}(s)+\tilde{\zeta}_{1}(-st)$

for $r>0$, and

$\displaystyle
ZP_{0}(se_{1})=\tilde{\zeta}_{1}(s)+\tilde{\zeta}_{1}(-st)+\zeta_{1}(s)+\zeta_{1}(-st).$

Similar to the proof in Lemma 23, we have $\zeta_{1}\equiv 0$. ∎

######  Theorem 31.

Let $n\geq 3$. A map $Z:\mathcal{P}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ is a continuous and $\mathrm{SL}(n)$
covariant valuation if and only if there are $\zeta,\tilde{\zeta}\in
C({\mathbb{R}})$ and $\mu,\tilde{\mu}\in\mathcal{M}^{c}({\mathbb{R}})$ such
that

$\displaystyle ZP(x)$
$\displaystyle=\zeta(h_{P}(x))+\zeta(-h_{-P}(x))+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(P\cap
H_{x,t})d\mu(t)$
$\displaystyle\qquad+\tilde{\zeta}(h_{[P,o]}(x))+\tilde{\zeta}(-h_{-[P,o]}(x))+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}([P,o]\cap
H_{x,t})d\tilde{\mu}(t)$

for every $P\in\mathcal{P}^{n}$ and $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

###### Proof.

We first prove that (55) holds for every $P\in\mathcal{P}^{n}$ and $\zeta\in
F({\mathbb{R}})$. By (7), (8) and (47), we only need to show that

$\displaystyle\mathcal{F}(P)\setminus\mathcal{F}^{-}(P)=\mathcal{F}([P,o])\setminus\mathcal{F}^{-}([P,o]).$

Let $F\in\mathcal{F}(P)\setminus\mathcal{F}^{-}(P)$ first. There is a
$u\in\operatorname{relint}N(P,F)$ such that $h_{P}(u)>0$. Hence $P\subset
H_{u,h_{P}(u)}^{-}$, $o\in\operatorname{relint}H_{u,h_{P}(u)}^{-}$ and
$F=P\cap H_{u,h_{P}(u)}$, where
$H_{u,h_{P}(u)}^{-}:=\\{x\in{\mathbb{R}}^{n}:x\cdot u\leq h_{P}(u)\\}$. Thus
$F\in\mathcal{F}([P,o])\setminus\mathcal{F}^{-}([P,o])$. Also, let
$F^{\prime}\in\mathcal{F}([P,o])\setminus\mathcal{F}^{-}([P,o])$. Similarly
there is a plane $H_{u,h_{[P,o]}(u)}$ that does not contain the origin and
$F^{\prime}=[P,o]\cap H_{u,h_{[P,o]}(u)}$. Hence
$F^{\prime}\in\mathcal{F}(P)\setminus\mathcal{F}^{-}(P)$. Thus (55) holds.

Lemma 9 shows that $ZP\in C({\mathbb{R}}^{n}\setminus\\{o\\})$ for every
$P\in\mathcal{P}^{n}$. Now applying Theorem 28 to $P\in\mathcal{P}_{o}^{n}$
with $\dim P<n$ and $P\in\mathcal{P}^{n}$ with $\dim P<n-1$ (by the proof of
Theorem 2 and Theorem 28, the assumption $ZP\in
C({\mathbb{R}}^{n}\setminus\\{o\\})$ is enough for this case), we get (30)
holds for every $P\in\mathcal{P}_{o}^{n}$ with $\dim P<n$ and for every
$P\in\mathcal{P}^{n}$ with $\dim P<n-1$. By the continuity of $Z$ and Lemma
30, we get $\zeta_{2}+\tilde{\zeta}_{2}\equiv 0$ and $\zeta_{1}\equiv 0$.

Combining (30) with (55), (7) and Corollary 24, there are suitable functions
$\zeta,\tilde{\zeta}\in C({\mathbb{R}})$ such that

$\displaystyle
ZP(x)=\zeta(h_{P}(x))+\zeta(-h_{-P}(x))+\tilde{\zeta}(h_{[P,o]}(x))+\tilde{\zeta}(-h_{-[P,o]}(x))$

for every $P\in\mathcal{P}^{n}$ if $\dim P<n-1$ and for every
$P\in\mathcal{P}_{o}^{n}$ if $\dim P<n$.

Define $\bar{Z}:\mathcal{P}^{n}\to F({\mathbb{R}}^{n}\setminus\\{o\\})$ by

$\displaystyle\bar{Z}P(x)=ZP(x)-(\zeta(h_{P}(x))+\zeta(-h_{-P}(x))+\tilde{\zeta}(h_{[P,o]}(x))+\tilde{\zeta}(-h_{-[P,o]}(x))).$

By Lemma 12, we only need to show that

$\displaystyle\bar{Z}P(x)=\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(P\cap
H_{x,t})d\mu(t)+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}([P,o]\cap
H_{x,t})d\tilde{\mu}(t)$

for all $P=sT^{n}$ and $P=s[e_{1},\dots,e_{n}]$ with $s>0$. By Lemma 9, the
remark after Theorem 13, and the same idea of the proofs of Theorem 28, we get
the desired result. ∎

Follows directly from Theorem 31 and conclusions in §4, we get the following.

######  Theorem 32.

Let $n\geq 3$. A map $Z:\mathcal{K}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ is a continuous and $\mathrm{SL}(n)$
covariant valuation if and only if there are $\zeta,\tilde{\zeta}\in
C({\mathbb{R}})$ and $\mu,\tilde{\mu}\in\mathcal{M}^{c}({\mathbb{R}})$ such
that

$\displaystyle ZK(x)$
$\displaystyle=\zeta(h_{K}(x))+\zeta(-h_{-K}(x))+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(K\cap
H_{x,t})d\mu(t)$
$\displaystyle\qquad+\tilde{\zeta}(h_{[K,o]}(x))+\tilde{\zeta}(-h_{-[K,o]}(x))+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}([K,o]\cap
H_{x,t})d\tilde{\mu}(t)$

for every $K\in\mathcal{K}^{n}$ and $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

##  11 Further results

In this section, we add some assumptions to Theorem 1 and Theorem 3 to get
more specific classifications and characterizations. Some results are new
(Corollaries 33, 36 and some part of Corollary 34), and some results are known
(Corollaries 35, 37, 38 and some part of Corollary 34).

A map $Z:\mathcal{P}_{o}^{n}\to F({\mathbb{R}}^{n}\setminus\\{o\\})$ is
$q$-homogeneous if $Z(\alpha P)=\alpha^{q}ZP$ for any $\alpha>0$ and
$P\in\mathcal{P}_{o}^{n}$. For $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$, the
half-space $H_{x,0}^{+}=\\{y\in{\mathbb{R}}^{n}:x\cdot y\geq 0\\}$ and
$H_{x,0}^{-}=\\{y\in{\mathbb{R}}^{n}:x\cdot y\leq 0\\}$.

######  Corollary 33.

Let $n\geq 3$ and $q\in{\mathbb{R}}$. A map $Z:\mathcal{P}_{o}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ is a continuous, $\mathrm{SL}(n)$
covariant and $q$-homogeneous valuation if and only if there are
$c_{1},c_{2},c_{3},c_{4}\in{\mathbb{R}}$ such that

$\displaystyle ZP(x)$
$\displaystyle=\begin{cases}c_{1}h_{P}(x)^{q}+c_{2}h_{-P}(x)^{q}+c_{3}\int_{P\cap
H_{x,o}^{+}}(x\cdot y)^{q-n}dy+c_{4}\int_{P\cap H_{x,o}^{-}}\left(-x\cdot
y\right)^{q-n}dy,&q>n-1,\\\ c_{1}h_{P}(x)^{q}+c_{2}h_{-P}(x)^{q},&0\leq q\leq
n-1,\\\ 0,&q<0\end{cases}$

for every $P\in\mathcal{P}_{o}^{n}$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

###### Proof.

Let $\zeta_{1}(t)$ be $c_{1}t^{q}$ for $t\geq 0$ and $c_{2}(-t)^{q}$ for $t<0$
($q>0$), and the density of $\mu$ be $c_{3}t^{q-n}$ for $t>0$ and
$c_{4}(-t)^{q-n}$ for $t<0$ ($q>n-1$) in Theorem 1 (and (2)). Hence the “if”
part is true.

For the “only if” part, by Theorem 1, there are $\zeta\in C({\mathbb{R}})$ and
$\mu\in\mathcal{M}^{c}({\mathbb{R}})$ such that

$\displaystyle ZP(x)$
$\displaystyle=\zeta(h_{P}(x))+\zeta(-h_{-P}(x))+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(P\cap
H_{x,t})d\mu(t)$

for every $P\in\mathcal{P}_{o}^{n}$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$. Since $Z$ is $q$-homogeneous, we have

$\displaystyle\zeta(h_{\alpha P}(x))+\zeta(-h_{-\alpha
P}(x))=\alpha^{q}(\zeta(h_{P}(x))+\zeta(-h_{-P}(x)))$

for $\dim P<n$, $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$ and $\alpha>0$. Further
setting $P=[o,e_{1}]$ and $x=e_{1}$ or $x=-e_{1}$, we get

$\displaystyle\zeta(\alpha)=\alpha^{q}\zeta(1),\zeta(-\alpha)=\alpha^{q}\zeta(-1).$

Let $c_{1}=\zeta(1)$ and $c_{2}=\zeta(-1)$. For $q\geq 0$, the condition
$\zeta\in C({\mathbb{R}})$ gives

$\displaystyle ZP(x)=c_{1}h_{P}(x)^{q}+c_{2}h_{-P}(x)^{q}$

for any $\dim P<n$, $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$. For $q<0$, the
condition $\zeta\in C({\mathbb{R}})$ gives $c_{1}=0$ and $c_{2}=0$.

Now let

$\displaystyle\tilde{Z}P(x)=\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(P\cap
H_{x,t})d\mu(t).$

By the above argument,
$\tilde{Z}P(x)=ZP(x)-c_{1}h_{P}(x)^{q}-c_{2}h_{-P}(x)^{q}$. Hence
$\tilde{Z}P(\cdot)$ is also $q$-homogeneous. Set $P=[0,1]\times[0,1]^{n-1}$
and $x=e_{1}$. Since $\tilde{Z}(\alpha P)(x)=\alpha^{q}\tilde{Z}P(x)$, we get

$\displaystyle\alpha^{n-1}\mu(0,\alpha)=\alpha^{q}\mu(0,1).$

For arbitrary $0<r<s$, together with $\mu\in\mathcal{M}^{c}({\mathbb{R}})$,
then we have

$\displaystyle\mu(r,s)=\mu(0,s)-\mu(0,r)=\mu(0,1)(s^{q-n+1}-r^{q-n+1})=\mu(0,1)(q-n+1)\int_{r}^{s}t^{q-n}dt$

Let $c_{3}=\mu(0,1)(q-n+1)$. Since the Radon measure $\mu$ on $(0,\infty)$ is
determined by its values on $(r,s)$ for all $0<r<s$ (regularity of the Radon
measure), $\mu$ has a density

$\displaystyle\zeta(t)=c_{3}t^{q-n}$

for $t>0$. Further since $\mu(0,s)<\infty$ for $s>0$, we have $c_{3}=0$ if
$q\leq n-1$. Similarly, change $x$ to $-e_{1}$, then $\mu$ has a density
$\zeta(t)=c_{4}t^{q-n}$ for $t<0$ with a constant $c_{4}\in{\mathbb{R}}$, and
we have $c_{4}=0$ if $q\leq n-1$. Now the proof is completed by (2). ∎

We can also assume that $ZP$ is $q$-homogeneous for every $P$. For
$q\in{\mathbb{R}}$, a function $f\in F({\mathbb{R}}^{n}\setminus\\{o\\})$ is
$q$-homogeneous if $f(\alpha x)=\alpha^{q}f(x)$ for any $\alpha>0$. The
following Corollary was established by Liu and Wang [31] for $q>-1$ such that
valuations are simple and Parapatits [43] for $q\geq 1$ for all continuous
valuations. The proof is quite similar to that of Corollary 34 (using the same
specific polytopes), so we omit the proof.

######  Corollary 34.

Let $n\geq 3$ and $q\in{\mathbb{R}}$. A map $Z$, mapping $\mathcal{P}_{o}^{n}$
to the space of $q$-homogeneous functions on
${\mathbb{R}}^{n}\setminus\\{o\\}$, is a continuous and $\mathrm{SL}(n)$
covariant valuation if and only if there are
$c_{1},c_{2},c_{3},c_{4}\in{\mathbb{R}}$ such that

$\displaystyle
ZP(x)=\begin{cases}c_{1}h_{P}(x)^{q}+c_{2}h_{-P}(x)^{q}+c_{3}\int_{P\cap
H_{x,o}^{+}}(x\cdot y)^{q}dy+c_{4}\int_{P\cap H_{x,o}^{-}}\left(-x\cdot
y\right)^{q}dy,&q\geq 0,\\\ c_{3}\int_{P\cap H_{x,o}^{+}}(x\cdot
y)^{q}dy+c_{4}\int_{P\cap H_{x,o}^{-}}\left(-x\cdot y\right)^{q}dy,&-1<q<0,\\\
0,&q\leq-1\end{cases}$

for every $P\in\mathcal{P}_{o}^{n}$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

A special case of $q=0$ in Corollary 34 is the classification of constant
function-valued valuations, which is equivalent with the classification of
$\mathrm{SL}(n)$ invariant real-valued valuations.

######  Corollary 35.

Let $n\geq 3$. A map $Z:\mathcal{P}_{o}^{n}\to{\mathbb{R}}$ is a continuous
and $\mathrm{SL}(n)$ invariant valuation if and only if there are
$c_{1},c_{2}\in{\mathbb{R}}$ such that

$\displaystyle ZP=c_{1}V_{0}(P)+c_{2}V_{n}(P)$

for every $P\in\mathcal{P}_{o}^{n}$.

###### Proof.

Set $\tilde{Z}P(x)=ZP$ for any $P\in\mathcal{P}_{o}^{n}$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$. Hence $\tilde{Z}$ is a continuous and
$\mathrm{SL}(n)$ covariant valuation. Also, $\tilde{Z}P$ is a $0$-homogeneous
function. By Corollary 34,

$\displaystyle\tilde{Z}P(x)=a_{1}+a_{2}+a_{3}V_{n}(P\cap
H_{x,o}^{+})+a_{4}V_{n}(P\cap H_{x,o}^{-})$

with some constants $a_{1},a_{2},a_{3},a_{4}\in{\mathbb{R}}$. Choosing an
asymmetric $P\in\mathcal{P}_{o}^{n}$, we get $a_{3}=a_{4}$ since
$\tilde{Z}P(x)=\tilde{Z}P(-x)$. Thus we get the desired result. ∎

The function $ZP(x)=\int_{P}\log|x\cdot y|dy$ is not homogeneous on $x$, but

$\displaystyle ZP(\alpha x)=ZP(x)+V(P)\log\alpha$

for any $P\in\mathcal{P}_{o}^{n}$, $\alpha>0$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

######  Corollary 36.

Let $n\geq 3$. A map $Z:\mathcal{P}_{o}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ is a continuous and $\mathrm{SL}(n)$
covariant valuation which satisfies the following property that

$\displaystyle ZP(\alpha x)=ZP(x)+\bar{Z}P(x)\log\alpha$ (57)

with some $\bar{Z}P(x)\in{\mathbb{R}}$ for any $P\in\mathcal{P}_{o}^{n}$,
$\alpha>0$ and $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$, if and only if there
are $c_{1},c_{2},c_{3},c_{4},c_{5}\in{\mathbb{R}}$ such that

$\displaystyle ZP(x)$ $\displaystyle=c_{1}V_{0}(P)+c_{2}V_{n}(P\cap
H_{x,o}^{+})+c_{3}V_{n}(P\cap H_{x,o}^{-})$
$\displaystyle\qquad+c_{4}\int_{P\cap H_{x,o}^{+}}\log|x\cdot
y|dy+c_{5}\int_{P\cap H_{x,o}^{-}}\log|x\cdot y|dy$

for every $P\in\mathcal{P}_{o}^{n}$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

###### Proof.

Let $\zeta_{1}\equiv c_{1}$ and the density of $\mu$ be $c_{2}+c_{4}\log|t|$
for $t>0$ and $c_{3}+c_{5}\log|t|$ for $t<0$ in Theorem 1 (and (2)). Also,
(57) holds for $\bar{Z}P(x)=c_{4}V_{n}(P\cap H_{x,o}^{+})+c_{5}V_{n}(P\cap
H_{x,o}^{-})$. Hence the “if” part is true.

Since $Z:\mathcal{P}_{o}^{n}\to F({\mathbb{R}}^{n}\setminus\\{o\\})$ is a
continuous, $\mathrm{SL}(n)$ covariant valuation, the property (57) implies
directly that $\bar{Z}:\mathcal{P}_{o}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ is also a continuous, $\mathrm{SL}(n)$
covariant valuation. Moreover, for any $\alpha,\beta>0$, (57) gives

$\displaystyle ZP(\alpha\beta x)=ZP(x)+\bar{Z}P(x)\log(\alpha\beta)$

and

$\displaystyle ZP(\alpha\beta x)=ZP(\alpha x)+\bar{Z}P(\alpha
x)\log\beta=ZP(x)+\bar{Z}P(x)\log\alpha+\bar{Z}P(\alpha x)\log\beta.$

Hence $\bar{Z}P(\cdot)$ is $0$-homogeneous. Now by Corollary 34 for $q=0$,
there are $b_{0},b_{1},b_{2}\in{\mathbb{R}}$ such that

$\displaystyle\bar{Z}P(x)=b_{0}V_{0}(P)+b_{1}V_{n}(P\cap
H_{x,o}^{+})+b_{2}V_{n}(P\cap H_{x,o}^{-}).$ (58)

for any $P\in\mathcal{P}_{o}^{n}$ and $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

Next we apply Theorem 1 and use a similar argument in Corollary 33 to prove
the “only if” part. By Theorem 1, there are $\zeta\in C({\mathbb{R}})$ and
$\mu\in\mathcal{M}^{c}({\mathbb{R}})$ such that

$\displaystyle ZP(x)$
$\displaystyle=\zeta(h_{P}(x))+\zeta(-h_{-P}(x))+\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(P\cap
H_{x,t})d\mu(t)$

Consider $P=[o,e_{1}]$ and $x=e_{1}$ or $x=-e_{1}$. By (57) and (58), we get

$\displaystyle\zeta(\alpha)=\zeta(1)+b_{0}\log\alpha,~{}\zeta(-\alpha)=\zeta(-1)+b_{0}\log\alpha$

for any $\alpha>0$. Recall that $\zeta\in C({\mathbb{R}})$. Thus $b_{0}=0$ and
$\zeta(\alpha)=\zeta(1)$ for any $\alpha\in{\mathbb{R}}$. Denoted by
$c_{1}=\zeta(1)$, we get

$\displaystyle ZP(x)=c_{1}V_{0}(P)$

for every $P\in\mathcal{P}_{o}^{n}$ with $\dim P<n$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$.

For the full-dimensional case, we set

$\displaystyle\tilde{Z}P(x)=\frac{1}{|x|}\int_{{\mathbb{R}}}V_{n-1}(P\cap
H_{x,t})d\mu(t),$

which is equal to $ZP(x)-c_{1}V_{0}(P)$ by the above argument. Set
$P=[0,1]\times[0,1]^{n-1}$ and $x=e_{1}$ or $x=-e_{1}$. By (57) and (58), we
have

$\displaystyle\alpha^{-1}\mu(0,\alpha)=\mu(0,1)+b_{1}\log\alpha,~{}~{}\alpha^{-1}\mu(-\alpha,0)=\mu(-1,0)+b_{2}\log\alpha.$

For arbitrary $0<r<s$, together with $\mu\in\mathcal{M}^{c}({\mathbb{R}})$, we
have

$\displaystyle\mu(r,s)=\mu(0,s)-\mu(0,r)$
$\displaystyle=\mu(0,1)(s-r)+b_{1}(s\log s-r\log r)$
$\displaystyle=\int_{r}^{s}\mu(0,1)+b_{1}+b_{1}\log t~{}dt$

and

$\displaystyle\mu(-s,-r)=\mu(-s,0)-\mu(-r,0)$
$\displaystyle=\mu(-1,0)(s-r)+b_{2}(s\log s-r\log r)$
$\displaystyle=\int_{-s}^{-r}\mu(-1,0)+b_{2}+b_{2}\log(-t)~{}dt$

Now set $c_{2}=\mu(0,1)+b_{1}$, $c_{3}=\mu(-1,0)+b_{2}$, $c_{4}=b_{1}$ and
$c_{5}=b_{2}$, then the proof is completed by (2). ∎

Corollary 34 has special geometric interests since it can be associated with
some geometric additions. The following two corollaries can be also found in
[17, 43, 31].

Let $\mathcal{K}_{c}^{n}$ be the set of origin-symmetric convex bodies in
${\mathbb{R}}^{n}$. A map $Z:\mathcal{P}_{o}^{n}\to\mathcal{K}_{c}^{n}$ is
$\mathrm{SL}(n)$ covariant if $Z(\phi P)=\phi ZP$ for any
$\phi\in\mathrm{SL}(n)$ and $P\in\mathcal{P}_{o}^{n}$.

######  Corollary 37.

Let $n\geq 3$ and $q\geq 1$. A map
$Z:\mathcal{P}_{o}^{n}\to\mathcal{K}_{c}^{n}$ is a continuous and
$\mathrm{SL}(n)$ covariant $L_{q}$ Minkowski valuation if and only if there
are $c_{1},c_{2}\in{\mathbb{R}}$ such that

$\displaystyle ZP(x)=c_{1}(P+_{q}(-P))+_{q}c_{2}M_{q}P$

for every $P\in\mathcal{P}_{o}^{n}$.

###### Proof.

Set $\tilde{Z}:\mathcal{P}_{o}^{n}\to F({\mathbb{R}}^{n}\setminus\\{o\\})$ by
$\tilde{Z}P(x)=h_{ZP}(x)^{q}$ for any $P\in\mathcal{P}_{o}^{n}$ and
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$. It is easy to see that $\tilde{Z}$ is
a continuous, $\mathrm{SL}(n)$ covariant valuation on $\mathcal{P}_{o}^{n}$
which takes values at $q$-homogeneous functions. Consider an asymmetric
$P\in\mathcal{P}_{o}^{n}$ for $\dim P<n$ and for $\dim P=n$ separately.
Together with (4), the desired result follows directly from Corollary 34. ∎

Let $\mathcal{S}_{c}^{n}$ denote the space of origin-symmetric, compact, star-
shaped sets which contain the origin in their interior. For $q\neq 0$, the
$L_{q}$ harmonic sum $K\widehat{+}_{q}L$ of $K,L\in\mathcal{S}_{c}^{n}$ is

$\displaystyle\|x\|_{K\widehat{+}_{q}L}=\left(\|x\|_{K}^{q}+\|x\|_{L}^{q}\right)^{1/q}.$

For $q\geq 1$, $K^{\ast}\widehat{+}_{q}L^{\ast}=(K+_{q}L)^{\ast}$.

We use the notation $\langle\mathcal{S}^{n}_{c},\widehat{+}_{q}\rangle$ as in
[16, 31] to denote the smallest monoid containing $\mathcal{S}^{n}_{c}$ with
respect to the $L_{q}$ harmonic addition. Thus
$\langle\mathcal{S}^{n}_{c},\widehat{+}_{q}\rangle$ contains the set
${\mathbb{R}}^{n}$ for $q>0$ and $\\{o\\}$ for $q<0$, respectively. A map
$Z:\mathcal{P}_{o}^{n}\to\langle\mathcal{S}^{n}_{c},\widehat{+}_{q}\rangle$ is
$\mathrm{SL}(n)$ contravariant if $Z(\phi P)=\phi^{-t}ZP$ for any
$\phi\in\mathrm{SL}(n)$ and $P\in\mathcal{P}_{o}^{n}$.

######  Corollary 38.

Let $n\geq 3$ and $q\neq 0$. A map
$Z:\mathcal{P}_{o}^{n}\to\langle\mathcal{S}^{n}_{c},\widehat{+}_{q}\rangle$ is
a continuous and $\mathrm{SL}(n)$ contravariant valuation if and only if there
is a constant $c_{1}\in{\mathbb{R}}$ such that

$\displaystyle ZP(x)=\begin{cases}c_{1}M_{q}^{\ast}P,&q>-1,\\\
\\{o\\},&q\leq-1.\end{cases}$

for every $P\in\mathcal{P}_{o}^{n}$.

###### Proof.

For $q\neq 0$, set $\tilde{Z}:\mathcal{P}_{o}^{n}\to
F({\mathbb{R}}^{n}\setminus\\{o\\})$ by $\tilde{Z}P(x)=\|x\|_{ZP}^{q}$ for any
$P\in\mathcal{P}_{o}^{n}$ and $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$. It is
easy to see that $\tilde{Z}$ is a continuous, $\mathrm{SL}(n)$ contravariant
valuation on $\mathcal{P}_{o}^{n}$ which takes values at $q$-homogeneous
functions. By Corollary 34, we have

$\displaystyle\|x\|_{ZP}^{q}=\begin{cases}c_{1}h_{P}(x)^{q}+c_{2}h_{-P}(x)^{q}+c_{3}\int_{P\cap
H_{x,o}^{+}}(x\cdot y)^{q}dy+c_{4}\int_{P\cap H_{x,o}^{-}}\left(-x\cdot
y\right)^{q}dy,&q>0,\\\ c_{3}\int_{P\cap H_{x,o}^{+}}(x\cdot
y)^{q}dy+c_{4}\int_{P\cap H_{x,o}^{-}}\left(-x\cdot y\right)^{q}dy,&-1<q<0,\\\
0,&q\leq-1.\end{cases}$

for $P\in\mathcal{P}_{o}^{n}$ and $x\in{\mathbb{R}}^{n}\setminus\\{o\\}$. Let
first $\dim P<n$. By the definition of
$\langle\mathcal{S}^{n}_{c},\widehat{+}_{q}\rangle$, either
$\|x\|_{ZP}^{q}\equiv 0$ or $\|x\|_{ZP}>0$ for all
$x\in{\mathbb{R}}^{n}\setminus\\{o\\}$. Hence $c_{1}=c_{2}=0$ (e.g. consider
$P=[o,e_{1}]$). Further consider $\dim P=n$ and an asymmetric $P$, then
$c_{3}=c_{4}$. Now by (5), we get the desired result for $q\neq 0$. (For
$q\leq-1$, $\|\cdot\|_{ZP}^{q}\equiv 0$ means $\|\cdot\|_{ZP}\equiv\infty$ and
hence $ZP=\\{o\\}$.) ∎

## Acknowledgement

The work of the author was supported in part by the Austrian Science Fund (FWF
M2642 and I3027), the European Research Council (ERC 306445), and the National
Natural Science Foundation of China (11671249).

## References

  * [1] J. Abardia and T. Wannerer, Aleksandrov-Fenchel inequalities for unitary valuations of degree $2$ and $3$, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1767–1791.
  * [2] S. Alesker, Continuous rotation invariant valuations on convex sets, Ann. of Math. 149 (1999), no. 3, 977–1005.
  * [3] S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture, Geom. Funct. Anal. 11 (2001), no. 2, 244–272.
  * [4] S. Alesker, Valuations on convex functions and convex sets and Monge-Ampère operators, Adv. Geom. 19 (2019), no. 3, 313–322.
  * [5] S. Alesker, A. Bernig and F. E. Schuster, Harmonic analysis of translation invariant valuations, Geom. Funct. Anal. 21 (2011), no. 4, 751–773.
  * [6] A. Bernig and J. H. G. Fu, Hermitian integral geometry, Ann. of Math. 173 (2011), no. 2, 907–945.
  * [7] K. J. Böröczky, E. Lutwak, D. Yang and G. Zhang, The log-Brunn-Minkowski inequality, Adv. Math. 231 (2012), 1974–1997.
  * [8] K. J. Böröczky, E. Lutwak, D. Yang and G. Zhang, The logarithmic minkowski problem, J. Amer. Math. Soc. 26 (2013), no. 3, 831–852.
  * [9] A. Colesanti, M. Ludwig and F. Mussnig, A homogeneous decomposition theorem for valuations on convex functions, J. Funct. Anal., in press.
  * [10] A. Colesanti, M. Ludwig and F. Mussnig, Minkowski valuations on convex functions, Calc. Var. Partial Differential Equations 56 (2017), no. 6, Art. 162, 29 pp.
  * [11] A. Colesanti, D. Pagnini, P. Tradacete and I. Villanueva, A class of invariant valuations on ${\rm Lip}(S^{n-1})$, Adv. Math. 366 (2020), 107069.
  * [12] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015, revised edition.
  * [13] R. J. Gardner, D. Hug and W. Weil, The Orlicz-Brunn-Minkowski theory: A general framework, additions, and inequalities, J. Differential. Geom. 97 (2014), no. 3, 427–476.
  * [14] E. Grinberg and G. Zhang, Convolutions, transforms, and convex bodies, Proc. London Math. Soc. (3) 78 (1999), no. 1, 77–115.
  * [15] C. Haberl, $L_{p}$ intersection bodies, Adv. Math. 217 (2008), no. 6, 2599–2624.
  * [16] C. Haberl, Star body valued valuations, Indiana Univ. Math. J. 58 (2009), no. 5, 2253–2276.
  * [17] C. Haberl, Minkowski valuations intertwining with the special linear group, J. Eur. Math. Soc. 14 (2012), no. 5, 1565–1597.
  * [18] C. Haberl and M. Ludwig, A characterization of $L_{p}$ intersection bodies, Int. Math. Res. Not. 10548 (2006), 1–29.
  * [19] C. Haberl and L. Parapatits, The centro-affine Hadwiger theorem, J. Amer. Math. Soc. 27 (2014), no. 3, 685–705.
  * [20] C. Haberl and L. Parapatits, Valuations and surface area measures, J. Reine Angew. Math. 687 (2014), 225–245.
  * [21] C. Haberl and F. E. Schuster, General $L_{p}$ affine isoperimetric inequalities, J. Differential Geom. 83 (2009), no. 1, 1–26.
  * [22] D. Hug and R. Schneider, Local tensor valuations, Geom. Funct. Anal. 24 (2014), no. 5, 1516–1564.
  * [23] K. Jochemko and R. Sanyal, Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem, J. Eur. Math. Soc. 20 (2018), no. 9, 2181–2208.
  * [24] Z. Kabluchko, G. Last and D. Zaporozhets, Inclusion-exclusion principles for convex hulls and the Euler relation, Discrete Comput. Geom. 58 (2017), no. 2, 417–434.
  * [25] D. Klain, A short proof of Hadwiger’s characterization theorem, Mathematika 42 (1995), no. 2, 329–339.
  * [26] B. Klartag, On convex perturbations with a bounded isotropic constant, Geom. Funct. Anal. 16 (2006), no. 6, 1271–1290.
  * [27] J. Li, Legendre transforms, Laplace transforms and valuations, in preparation.
  * [28] J. Li, Affine function-valued valuations, Int. Math. Res. Not. (2018), in press, Article ID: rny212.
  * [29] J. Li and G. Leng, $L_{p}$ Minkowski valuations on polytopes, Adv. Math. 299 (2016), 139–173.
  * [30] J. Li and D. Ma, Laplace transforms and valuations, J. Funct. Anal. 272 (2017), no. 2, 738–758.
  * [31] L. Liu and W. Wang, $SL(n)$ Contravariant $L_{p}$ Harmonic Valuations on Polytopes, Discrete Comput. Geom. (2020), in press.
  * [32] M. Ludwig, Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4191–4213.
  * [33] M. Ludwig, Intersection bodies and valuations, Amer. J. Math. 128 (2006), no. 6, 1409–1428.
  * [34] M. Ludwig, Valuations on Sobolev spaces, Amer. J. Math. 134 (2012), 824–842.
  * [35] M. Ludwig and M. Reitzner, A classification of SL$(n)$ invariant valuations, Ann. of Math. 172 (2010), no. 2, 1223–1271.
  * [36] M. Ludwig and M. Reitzner, SL$(n)$ invariant valuations on polytopes, Discrete Comput. Geom. 57 (2017), no. 3, 571–581.
  * [37] E. Lutwak, D. Yang and G. Zhang, $L_{p}$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111–132.
  * [38] E. Lutwak, D. Yang and G. Zhang, A new ellipsoid associated with convex bodies, Duke. Math. J. 104 (2000), no. 3, 375–390.
  * [39] E. Lutwak, D. Yang and G. Zhang, Orlicz centroid bodies, J. Differential Geom. 84 (2010), no. 2, 365–387.
  * [40] E. Lutwak, D. Yang and G. Zhang, Orlicz projection bodies, Adv. Math. 223 (2010), no. 1, 220–242.
  * [41] P. McMullen, Valuations and Euler-type relations on certain classes of convex polytopes, Proc. London Math. Soc. 35 (1977), no. 1, 113–135.
  * [42] F. Mussnig, Volume, polar volume and Euler characteristic for convex functions, Adv. Math. 344 (2019), 340–373.
  * [43] L. Parapatits, SL(n)-covariant $L_{p}$-Minkowski valuations, J. London Math. Soc. 89 (2014), no. 2, 397–414.
  * [44] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 2014, 2nd edition.
  * [45] F. E. Schuster, Crofton measures and Minkowski valuations, Duke Math. J. 154 (2010), no. 1, 1–30.
  * [46] F. E. Schuster and T. Wannerer, Even Minkowski valuations, Amer. J. Math. 137 (2015), no. 6, 1651–1683.
  * [47] F. E. Schuster and T. Wannerer, Minkowski valuations and generalized valuations, J. Eur. Math. Soc. 20 (2018), no. 8, 1851–1884.
  * [48] G. C. Shephard, Euler-type relations for convex polytopes, Proc. London Math. Soc. (3) 18 (1968), 597–606.
  * [49] T. Wannerer, GL(n) equivariant Minkowski valuations, Indiana Univ. Math. J. 60 (2011), no. 5, 1655–1672.
  * [50] D. Xi, H. Jin and G. Leng, The Orlicz Brunn-Minkowski inequality, Adv. Math. 260 (2014), 350–374.
  * [51] V. Yaskin and M. Yaskina, Centroid bodies and comparison of volumes, Indiana Univ. Math. J. 55 (2006), no. 3, 1175–1194.