Valuations on Convex Functions

TL;DR

This paper classifies all continuous, SL(n), and translation invariant valuations on convex functions in b^n, providing a complete understanding of their structure and properties.

Contribution

It offers a complete classification of valuations on convex functions that are continuous, SL(n) invariant, and translation invariant, filling a gap in valuation theory.

Findings

Complete classification of valuations on convex functions.

Characterization of valuations invariant under SL(n) and translations.

Framework for understanding valuations in convex analysis.

Abstract

All continuous, SL$(n)$ and translation invariant valuations on the space of convex functions on ${\mathbb R}^n$ are completely classified.

Equations

$$Z(u\vee v)+Z(u\wedge v)=Z(u)+Z(v)$$

$$\underset{|x|\to +\infty }{\lim }u(x)=+\infty$$

$$\operatorname{Z}(u)=\zeta_{0}\big{(}\min\nolimits_{x\in\mathbb{R}^{n}}u(x)\big{)}+\int_{\operatorname{dom}u}\zeta_{n}\big{(}u(x)\big{)}\,dx$$

$$Z(P)=c_0{V}_0(P)+c_n{V}_n(P)$$

$$\{u<t\}=\{x\in {\mathbb{R}}^n:u(x)<t\},\{u\le t\}=\{x\in {\mathbb{R}}^n:u(x)\le t\},$$

$$\mathrm{epi}u=\{(x,y)\in {\mathbb{R}}^n\times \mathbb{R}:u(x)\le y\},$$

$$\{u\wedge v\le t\}=\{u\le t\}\cup \{v\le t\},\{u\vee v\le t\}=\{u\le t\}\cap \{v\le t\},$$

$$\mathrm{relint}\{u\le t\}\subseteq \{u<t\}$$

$$u(x)\le \underset{k\to \infty }{\mathrm{liminf}}{u}_k(x_k).$$

$$u(x)=\underset{k\to \infty }{\lim }{u}_k(x_k).$$

$${epi-lim}_{k\to \infty }{u}_k(x)\le {\lim }_{k\to \infty }{u}_k(x),$$

$$K+L=\{x+y:x\in K,y\in L\}$$

$$\delta (K,L)=\inf \{\epsilon >0:K\subset L+\epsilon B,L\subset K+\epsilon B\}.$$

$$u_k(x_k)\le s+\epsilon \le t.$$

$$u(x)\le \underset{j\to \infty }{\mathrm{liminf}}{u}_{i_j}(x_{i_j})\le t$$

$$u_k(x)>t.$$

$$x_{i_j}\in \{u_{i_j}\le t\}\subseteq \{u_{i_j}\le u_{\min }+1\}.$$

$$u(x)\le \underset{k\to \infty }{\mathrm{liminf}}{u}_{i_{j_k}}(x_{i_{j_k}})\le t.$$

$$u^{*}(y)=\sup_{x\in\mathbb{R}^{n}}\big{(}\langle y,x\rangle-u(x)\big{)},\quad y\in\mathbb{R}^{n}.$$

$$(u_{1}\mathbin{\Box}u_{2})(x)=\inf_{x=x_{1}+x_{2}}\big{(}u_{1}(x_{1})+u_{2}(x_{2})\big{)},\quad x\in\mathbb{R}^{n}.$$

$$\mathrm{epi}(u_1\square u_2)=\mathrm{epi}{u}_1+\mathrm{epi}{u}_2.$$

$$(u_1\square u_2{)}^{\ast }=u_1^{\ast }+u_2^{\ast }.$$

$$u_t(x)=tu\left(\frac{x}{t}\right).$$

$$u_{t}^{*}(y)=\sup_{x\in\mathbb{R}^{n}}\big{(}\langle y,x\rangle-t\,u\left(\tfrac{x}{t}\right)\big{)}=\sup_{x\in\mathbb{R}^{n}}\big{(}\langle y,tx\rangle-t\,u(x)\big{)}=t\,u^{*}(y)$$

$$u_k\stackrel{epi}{⟶}u⟺u_k^{\ast }\stackrel{epi}{⟶}{u}^{\ast }.$$

$$u(x)>a|x|+b$$

$$u_k(x)>a|x|+b\text{ and }u(x)>a|x|+b$$

$$u(x)>c|x|+d=l(x).$$

$$l^{*}(y)=\Big{(}\!\sup_{x\in\mathbb{R}^{n}}\big{(}\langle y,x\rangle-c\,|x|\big{)}\Big{)}-d$$

$$\sup_{x\in\mathbb{R}^{n}}\big{(}\langle y,x\rangle-c\,|x|\big{)}=\begin{cases}0&\mbox{if }|y|\leq c\\ +\infty&\mbox{if }|y|>c.\end{cases}$$

Full text

2000 AMS subject classification: 26B25 (46A40, 52A20, 52B45).

A function $\operatorname{Z}$ defined on a lattice $({\mathcal{L}},\vee,\wedge)$ and taking values in an abelian semigroup is called a valuation if

[TABLE]

for all $u,v\in{\mathcal{L}}$. A function $\operatorname{Z}$ defined on some subset ${\mathcal{S}}$ of ${\mathcal{L}}$ is called a valuation on ${\mathcal{S}}$ if (1) holds whenever $u,v,u\vee v,u\wedge v\in{\mathcal{S}}$. For ${\mathcal{S}}$ the set of compact convex sets, ${\mathcal{K}}^{n}$, in $\mathbb{R}^{n}$ with $\vee$ denoting union and $\wedge$ intersection, valuations have been studied since Dehn’s solution of Hilbert’s Third Problem in 1901 and interesting new ones keep arising (see, for example, [16]). The natural topology on ${\mathcal{K}}^{n}$ is induced by the Hausdorff metric and continuous, $\operatorname{SL}(n)$ and translation invariant valuations on ${\mathcal{K}}^{n}$ were first classified by Blaschke. The celebrated Hadwiger classification theorem establishes a complete classification of continuous, rigid motion invariant valuations on ${\mathcal{K}}^{n}$ and provides a characterization of intrinsic volumes. See [2, 3, 6, 12, 25, 14, 13, 20, 1] for some recent results on valuations on convex sets and [15, 17] for information on the classical theory.

More recently, valuations have been studied on function spaces. Here ${\mathcal{S}}$ is a space of real valued functions and $u\vee v$ is the pointwise maximum of $u$ and $v$ while $u\wedge v$ is the pointwise minimum. For Sobolev spaces [23, 21, 27] and $L^{p}$ spaces [33, 32, 24] complete classifications for valuations intertwining the $\operatorname{SL}(n)$ were established. See also [22, 19, 34, 29]. Moreover, classical functionals for convex sets including the intrinsic volumes have been extended to the space of quasi-concave functions in [7] and [28] (see also [9, 18]). A classification of rigid motion invariant valuations on quasi-concave functions is established in [10]. For definable functions such a result was previously established in [5].

The aim of this paper is to establish a complete classification of $\operatorname{SL}(n)$ and translation invariant valuations on convex functions. Let $\operatorname{Conv}(\mathbb{R}^{n})$ denote the space of convex functions $u:\mathbb{R}^{n}\to(-\infty,+\infty]$ which are proper, lower semicontinuous and coercive. Here a function is proper if it is not identically $+\infty$ and it is coercive if

[TABLE]

where $\lvert x\rvert$ is the Euclidean norm of $x$. The space $\operatorname{Conv}(\mathbb{R}^{n})$ is one of the standard spaces in convex analysis and it is equipped with the topology associated to epi-convergence (see Section 1).

Let $n\geq 2$ throughout the paper. A functional $\operatorname{Z}:\operatorname{Conv}(\mathbb{R}^{n})\to\mathbb{R}$ is $\operatorname{SL}(n)$ ​invariant if $\operatorname{Z}(u\circ\phi^{-1})=\operatorname{Z}(u)$ for every $u\in\operatorname{Conv}(\mathbb{R}^{n})$ and $\phi\in\operatorname{SL}(n)$. It is translation invariant if $\operatorname{Z}(u\circ\tau^{-1})=\operatorname{Z}(u)$ for every $u\in\operatorname{Conv}(\mathbb{R}^{n})$ and translation $\tau:\mathbb{R}^{n}\to\mathbb{R}^{n}$. In [8], a class of rigid motion invariant valuations on $\operatorname{Conv}(\mathbb{R}^{n})$ was introduced and classification results were established. However, the setting is different from our setting, as a different topology (coming from a notion of monotone convergence) is used in [8] and monotonicity of the valuations is assumed. Variants of the functionals from [8] also appear in our classification. We say that a functional $\operatorname{Z}:\operatorname{Conv}(\mathbb{R}^{n})\to\mathbb{R}$ is continuous if $\operatorname{Z}(u)=\lim_{k\to\infty}\operatorname{Z}(u_{k})$ for every sequence $u_{k}\in\operatorname{Conv}(\mathbb{R}^{n})$ that epi-converges to $u\in\operatorname{Conv}(\mathbb{R}^{n})$.

[TABLE]

for every $u\in\operatorname{Conv}(\mathbb{R}^{n})$.

Here, a function $\zeta:\mathbb{R}\to[0,\infty)$ has finite $(n-1)$-st moment if $\int_{0}^{+\infty}t^{n-1}\zeta(t)\,\mathrm{d}t<+\infty$ and $\operatorname{dom}u$ is the domain of $u$, that is, $\operatorname{dom}u=\{x\in\mathbb{R}^{n}:u(x)<+\infty\}$. Since $u\in\operatorname{Conv}(\mathbb{R}^{n})$, the minimum of $u$ is attained and hence finite.

If the valuation in (3) is evaluated for a (convex) indicator function $\mathrm{I}_{K}$ for $K\in{\mathcal{K}}^{n}$ (where $\mathrm{I}_{K}(x)=0$ for $x\in K$ and $\mathrm{I}_{K}(x)=+\infty$ for $x\not\in K$), then $\zeta_{0}(0)V_{0}(K)+\zeta_{n}(0)\,V_{n}(K)$ is obtained, where $V_{0}(K)$ is the Euler characteristic and $V_{n}(K)$ the $n$-dimensional volume of $K$. The proof of the theorem makes essential use of the following classification of continuous and $\operatorname{SL}(n)$ invariant valuations on ${\mathcal{P}}^{n}_{0}$, the space of convex polytopes which contain the origin. A functional $\operatorname{Z}:{\mathcal{P}}^{n}_{0}\to\mathbb{R}$ is a continuous and $\operatorname{SL}(n)$ invariant valuation if and only if there are constants $c_{0},c_{n}\in\mathbb{R}$ such that

[TABLE]

for every $P\in{\mathcal{P}}^{n}_{0}$ (see, for example, [26]). For continuous and rotation invariant valuations on ${\mathcal{K}}^{n}$ that have polynomial behavior with respect to translations, a classification was established by Alesker [2] but a classification of continuous and rotation invariant valuations on ${\mathcal{P}}_{0}^{n}$ is not known. It is also an open problem to establish a classification of continuous and rigid motion invariant valuations on $\operatorname{Conv}(\mathbb{R}^{n})$.

1 The Space of Convex Functions

We collect some properties of convex functions and of the space $\operatorname{Conv}(\mathbb{R}^{n})$. A basic reference is the book by Rockafellar & Wets [30] (see also [11, 4]). In particular, epi-convergence is discussed and some properties of epi-convergent sequences of convex functions are established. For these results, conjugate functions are introduced. We also discuss piecewise affine functions and give a self-contained proof that they are dense in $\operatorname{Conv}(\mathbb{R}^{n})$.

To every convex function $u:\mathbb{R}^{n}\to(-\infty,+\infty]$, there can be assigned several convex sets. For $t\in(-\infty,+\infty]$, the sublevel sets

[TABLE]

are convex. The domain, $\operatorname{dom}u$, of $u$ is convex and the epigraph of $u$,

[TABLE]

is a convex subset of $\mathbb{R}^{n}\times\mathbb{R}$.

The lower semicontinuity of a convex function $u:\mathbb{R}^{n}\to(-\infty,+\infty]$ is equivalent to its epigraph being closed and to all sublevel sets, $\{u\leq t\}$, being closed. Such functions are also called closed. The growth condition (2) is equivalent to the boundedness of all sublevel sets $\{u\leq t\}$. Hence, $\{u\leq t\}\in{\mathcal{K}}^{n}$ for $u\in\operatorname{Conv}(\mathbb{R}^{n})$ and $t\geq\min_{x\in\mathbb{R}^{n}}u(x)$.

For convex functions $u,v\in\operatorname{Conv}(\mathbb{R}^{n})$, the pointwise minimum $u\wedge v$ corresponds to the union of their epigraphs and therefore to the union of their sublevel sets. Similarly, the pointwise maximum $u\vee v$ corresponds to the intersection of the epigraphs and sublevel sets. Hence for all $t\in\mathbb{R}$

[TABLE]

where for $u\vee v\in\operatorname{Conv}(\mathbb{R}^{n})$ all sublevel sets are either empty or in ${\mathcal{K}}^{n}$. For $u\in\operatorname{Conv}(\mathbb{R}^{n})$,

[TABLE]

for every $t>\min_{x\in\mathbb{R}^{n}}u(x)$, where $\operatorname{relint}$ is the relative interior (see [8, Lemma 3.2]).

1.1 Epi-convergence

A sequence $u_{k}:\mathbb{R}^{n}\to(-\infty,\infty]$ is epi-convergent to $u:\mathbb{R}^{n}\to(-\infty,\infty]$ if for all $x\in\mathbb{R}^{n}$ the following conditions hold:

• (i)

For every sequence $x_{k}$ that converges to $x$,

[TABLE]

• (ii)

There exists a sequence $x_{k}$ that converges to $x$ such that

[TABLE]

In this case we also write $u=\operatorname{epi-lim}_{k\to\infty}u_{k}$ and $u_{k}\stackrel{{\scriptstyle epi}}{{\longrightarrow}}u$.

Equation (6) means, that $u$ is an asymptotic common lower bound for the sequence $u_{k}$. Consequently, (7) states that this bound is optimal. The name epi-convergence is due to the fact, that this convergence is equivalent to the convergence of the corresponding epigraphs in the Painlevé-Kuratowski sense. Another name for epi-convergence is $\Gamma$-convergence (see [11, Theorem 4.16] and [30, Proposition 7.2]). We consider $\operatorname{Conv}(\mathbb{R}^{n})$ with the topology associated to epi-convergence.

Immediately from the definition of epi-convergence we get the following result (see, for example, [11, Proposition 6.1.]).

Lemma 1**.**

If $u_{k}:\mathbb{R}^{n}\to(-\infty,\infty]$ is a sequence that epi-converges to $u:\mathbb{R}^{n}\to(-\infty,\infty]$, then also every subsequence $u_{k_{i}}$ of $u_{k}$ epi-converges to $u$.

For the following result, see, for example, [30, Proposition 7.4 and Theorem 7.17].

Lemma 2**.**

If $u_{k}$ is a sequence of convex functions that epi-converges to a function $u$, then $u$ is convex and lower semicontinuous. Moreover, if $\operatorname{dom}u$ has non-empty interior, then $u_{k}$ converges uniformly to $u$ on every compact set that does not contain a boundary point of $\operatorname{dom}u$.

We also require the following connection to pointwise convergence (see, for example, [11, Example 5.13]).

Lemma 3**.**

Let $u_{k}:\mathbb{R}^{n}\to\mathbb{R}$ be a sequence of finite convex functions and $u:\mathbb{R}^{n}\to\mathbb{R}$ a finite convex function. Then $u_{k}$ is epi-convergent to $u$, if and only if $\,u_{k}$ converges pointwise to $u$.

The last statement is no longer true if the functions may attain the value $+\infty$. In that case

[TABLE]

for all $x\in\mathbb{R}^{n}$ such that both limits exist.

We want to connect epi-convergence of functions from $\operatorname{Conv}(\mathbb{R}^{n})$ with the convergence of their sublevel sets. The natural topology on ${\mathcal{K}}^{n}$ is induced by the Hausdorff distance. For $K,L\subset\mathbb{R}^{n}$, we write

[TABLE]

for their Minkowski sum. Let $B\subset\mathbb{R}^{n}$ be the closed, $n$-dimensional unit ball. For $K,L\in{\mathcal{K}}^{n}$, the Hausdorff distance is

[TABLE]

We write $K_{i}\to K$ as $i\to\infty$, if $\delta(K_{i},K)\to 0$ as $i\to\infty$. For the next result we need the following description of Hausdorff convergence on ${\mathcal{K}}^{n}$ (see, for example, [31, Theorem 1.8.8]).

Lemma 4**.**

The convergence $\lim_{i\to\infty}K_{i}=K$ in ${\mathcal{K}}^{n}$ is equivalent to the following conditions taken together:

1. (i)

Each point in $K$ is the limit of a sequence $(x_{i})_{i\in\mathbb{N}}$ with $x_{i}\in K_{i}$ for $i\in\mathbb{N}$. 2. (ii)

The limit of any convergent sequence $(x_{i_{j}})_{j\in\mathbb{N}}$ with $x_{i_{j}}\in K_{i_{j}}$ for $j\in\mathbb{N}$ belongs to $K$.

Each sublevel set of a function from $\operatorname{Conv}(\mathbb{R}^{n})$ is either empty or in ${\mathcal{K}}^{n}$. We say that $\{u_{k}\leq t\}\to\emptyset$ as $k\to\infty$ if there exists $k_{0}\in\mathbb{N}$ such that $\{u_{k}\leq t\}=\emptyset$ for $k\geq k_{0}$. We include the proof of the following simple result, for which we did not find a suitable reference.

Lemma 5**.**

Let $u_{k},u\in\operatorname{Conv}(\mathbb{R}^{n})$. If $u_{k}\stackrel{{\scriptstyle epi}}{{\longrightarrow}}u$ as $k\to\infty$, then $\{u_{k}\leq t\}{\to}\{u\leq t\}$ as $k\to\infty$ for every $t\in\mathbb{R}$ with $t\neq\min_{x\in\mathbb{R}^{n}}u(x)$.

Proof.

First, let $t>\min_{x\in\mathbb{R}^{n}}u(x)$. For $x\in\operatorname{relint}\{u\leq t\}$, it follows from (5) that $s=u(x)<t$. Since $u_{k}$ epi-converges to $u$, there exists a sequence $x_{k}$ that converges to $x$ such that $u_{k}(x_{k})$ converges to $u(x)$. Therefore, there exist $\varepsilon>0$ and $k_{0}\in\mathbb{N}$ such that for all $k\geq k_{0}$

[TABLE]

Thus, $x_{k}\in\{u_{k}\leq t\}$, which shows that $x$ is a limit of a sequence of points from $\{u_{k}\leq t\}$. It is easy to see that this implies $(i)$ of Lemma 4.

Now, let $(x_{i_{j}})_{j\in\mathbb{N}}$ be a convergent sequence in $\{u_{i_{j}}\leq t\}$ with limit $x\in\mathbb{R}^{n}$. By Lemma 1, the subsequence $u_{i_{j}}$ epi-converges to $u$. Therefore

[TABLE]

which gives (ii) of Lemma 4.

Second, let $t<\min_{x\in\mathbb{R}^{n}}u(x)=u_{\min}$. Since $\{u\leq t\}=\emptyset$, we have to show that there exists $k_{0}\in\mathbb{N}$ such that for every $k\geq k_{0}$ and $x\in\mathbb{R}^{n}$,

[TABLE]

Assume that there does not exist such an index $k_{0}$. Then there are infinitely many points $x_{i_{j}}$ such that $u_{i_{j}}(x_{i_{j}})\leq t$. Note, that

[TABLE]

By Lemma 1, we know that $u_{i_{j}}\stackrel{{\scriptstyle epi}}{{\longrightarrow}}u$ and therefore we can apply the previous argument to obtain that $\{u_{i_{j}}\leq u_{\min}+1\}\to\{u\leq u_{\min}+1\}$, which shows that the sequence $x_{i_{j}}$ is bounded. Hence, there exists a convergent subsequence $x_{i_{j_{k}}}$ with limit $x\in\mathbb{R}^{n}$. Applying Lemma 1 again, we obtain that $u_{i_{j_{k}}}$ is epi-convergent to $u$ and therefore

[TABLE]

This is a contradiction. Hence $\{u_{k}\leq t\}$ must be empty eventually. ∎

1.2 Conjugate functions and the cone property

We require a uniform lower bound for an epi-convergent sequence of functions from $\operatorname{Conv}(\mathbb{R}^{n})$. This is established by showing that all epigraphs are contained in a suitable cone that is given by the function $a\,|x|+b$ with $a>0$ and $b\in\mathbb{R}$. To establish this uniform cone property of an epi-convergent sequence, we use conjugate functions.

For a convex function $u:\mathbb{R}^{n}\to(-\infty,+\infty]$, its conjugate function $u^{*}:\mathbb{R}^{n}\to(-\infty,+\infty]$ is defined as

[TABLE]

Here $\langle y,x\rangle$ is the inner product of $x,y\in\mathbb{R}^{n}$. If $u$ is a closed convex function, then also $u^{*}$ is a closed convex function and $u^{**}=u$. Conjugation reverses inequalities, that is, if $u\leq v$, then $u^{*}\geq v^{*}$.

The infimal convolution of two closed convex functions $u_{1},u_{2}:\mathbb{R}^{n}\to(-\infty,+\infty]$ is defined by

[TABLE]

This just corresponds to the Minkowski addition of the epigraphs of $u_{1}$ and $u_{2}$, that is

[TABLE]

We remark that for two closed convex functions $u_{1},u_{2}$ the infimal convolution $u_{1}\mathbin{\Box}u_{2}$ need not be closed, even when it is convex. If $u_{1}\Box u_{2}>-\infty$ pointwise, then

[TABLE]

For $t>0$ and a closed convex function $u$ define the function $u_{t}$ by

[TABLE]

For the convex conjugate of $u_{t}$, we have

[TABLE]

(see, for example, [31], Section 1.6.2).

The next result shows a fundamental relationship between convex functions and their conjugates. It was first established by Wijsman (see [30, Theorem 11.34]).

Lemma 6**.**

If $u_{k},u\in\operatorname{Conv}(\mathbb{R}^{n})$, then

[TABLE]

The cone property was established in [9, Lemma 2.5] for functions in $\operatorname{Conv}(\mathbb{R}^{n})$.

Lemma 7**.**

For $u\in\operatorname{Conv}(\mathbb{R}^{n})$, there exist constants $a,b\in\mathbb{R}$ with $a>0$ such that

[TABLE]

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

Next, we extend this result to an epi-convergent sequence of functions in $\operatorname{Conv}(\mathbb{R}^{n})$ and obtain a uniform cone property.

Lemma 8**.**

Let $u_{k},u\in\operatorname{Conv}(\mathbb{R}^{n})$. If $u_{k}\stackrel{{\scriptstyle epi}}{{\longrightarrow}}u$, then there exist constants $a,b\in\mathbb{R}$ with $a>0$ such that

[TABLE]

for every $k\in\mathbb{N}$ and $x\in\mathbb{R}^{n}$.

Proof.

By Lemma 7, there exist constants $c>0$ and $d$ such that

[TABLE]

Switching to conjugates gives $u^{*}<l^{*}$. Note that

[TABLE]

and

[TABLE]

Hence $l^{*}=\mathrm{I}_{cB}-d$, where $cB$ is the closed centered ball with radius $c$. Set $a=c/2>0$. Hence $aB$ is a compact subset of $\operatorname{int}\operatorname{dom}u^{*}$. Therefore, Lemma 6 and Lemma 2 imply that $u_{k}^{*}$ converges uniformly to $u^{*}$ on $aB$. Since $u^{*}<-d$ on $aB$, there exists a constant $b$ such that $u_{k}^{*}(y)<-b$ for every $y\in aB$ and $k\in\mathbb{N}$ and therefore

[TABLE]

for every $k\in\mathbb{N}$. Consequently

[TABLE]

for every $k\in\mathbb{N}$ and $x\in\mathbb{R}^{n}$. ∎

Note, that Lemma 5 and Lemma 8 are no longer true if $u\equiv{+\infty}$. For example, consider $u_{k}(x)=\mathrm{I}_{k\,r(B+k^{2}x_{0})}$ for some $r>0$ and $x_{0}\in\mathbb{R}^{n}\backslash\{0\}$. Then $u_{k}$ epi-converges to $u$ but every set $\{u_{k}\leq t\}$ for $t\geq 0$ is a ball of radius $kr$. In this case, the sublevel sets are not even bounded. Moreover, it is clear that there does not exist a uniform pointed cone that contains all the sets $\operatorname{epi}u_{k}$.

1.3 Piecewise affine functions

A polyhedron is the intersection of finitely many closed halfspaces. A function $u\in\operatorname{Conv}(\mathbb{R}^{n})$ is called piecewise affine, if there exist finitely many $n$-dimensional convex polyhedra $C_{1},\ldots,C_{m}$ with pairwise disjoint interiors such that $\bigcup_{i=1}^{m}C_{i}=\mathbb{R}^{n}$ and the restriction of $u$ to each $C_{i}$ is an affine function. The set of piecewise affine and convex functions will be denoted by $\operatorname{Conv_{p.a.}}(\mathbb{R}^{n})$. We call $u\in\operatorname{Conv}(\mathbb{R}^{n})$ a finite element of $\operatorname{Conv}(\mathbb{R}^{n})$ if $u(x)<+\infty$ for every $x\in\mathbb{R}$. Note that piecewise affine and convex functions are finite elements of $\operatorname{Conv}(\mathbb{R}^{n})$. We want to show that $\operatorname{Conv_{p.a.}}(\mathbb{R}^{n})$ is dense in $\operatorname{Conv}(\mathbb{R}^{n})$ and use the Moreau-Yosida approximation in our proof. That $\operatorname{Conv_{p.a.}}(\mathbb{R}^{n})$ is dense in $\operatorname{Conv}(\mathbb{R}^{n})$ can also be deduced from more general results (see, [4, Corollary 3.42]).

Let $u\in\operatorname{Conv}(\mathbb{R}^{n})$ and $t>0$. Set $q(x)=\tfrac{1}{2}|x|^{2}$ and recall that $q_{t}(x)=\frac{t}{2}\,q(\frac{x}{t})$. The Moreau-Yosida approximation of $u$ is defined as

[TABLE]

or equivalently

[TABLE]

See, for example, [30, Chapter 1, Section G]. We require the following simple properties of the Moreau-Yosida approximation.

Lemma 9**.**

For $u\in\operatorname{Conv}(\mathbb{R}^{n})$, the Moreau-Yosida approximation $e_{t}u$ is a finite element of $\operatorname{Conv}(\mathbb{R}^{n})$ for every $t>0$. Moreover, $e_{t}u(x)\leq u(x)$ for $x\in\mathbb{R}^{n}$ and $t>0$.

Proof.

Fix $t>0$. Since

[TABLE]

for $x\in\mathbb{R}^{n}$, we have $e_{t}u(x)\leq u(x)$ for all $x\in\mathbb{R}^{n}$. Since $u$ is proper, there exists $x_{0}\in\mathbb{R}^{n}$ such that $u(x_{0})<+\infty$. This shows that

[TABLE]

for $x\in\mathbb{R}^{n}$, which shows that $e_{t}u$ is finite. Using (8) we obtain that

[TABLE]

It is therefore easy to see, that $e_{t}u$ is a convex function such that $\lim_{|x|\to+\infty}e_{t}u(x)=+\infty$. ∎

Lemma 10**.**

For every $u\in\operatorname{Conv}(\mathbb{R}^{n})$, $\,\operatorname{epi-lim}_{t\to 0^{+}}e_{t}u=u$.

Proof.

By Lemma 6, we have $e_{t}u\stackrel{{\scriptstyle epi}}{{\longrightarrow}}u$ if and only if $(e_{t}u)^{*}\stackrel{{\scriptstyle epi}}{{\longrightarrow}}u^{*}$. By the definition of $e_{t}$, (9) and (10), we have

[TABLE]

Therefore, we need to show that $u^{*}+tq^{*}\stackrel{{\scriptstyle epi}}{{\longrightarrow}}u^{*}$. For $q(x)=\tfrac{1}{2}|x|^{2}$, we have $q=q^{*}$. Since epi-convergence is equivalent to pointwise convergent if the functions are finite, it follows that $\operatorname{epi-lim}_{t\to 0^{+}}tq^{*}={0}$. It is now easy to see that \operatorname{epi-lim}_{t\to 0^{+}}\big{(}u^{*}+tq^{*})=u^{*} and therefore $\operatorname{epi-lim}_{t\to 0^{+}}(e_{t}u)^{*}=u^{*}$. ∎

Lemma 11**.**

$\operatorname{Conv_{p.a.}}(\mathbb{R}^{n})$* is dense in $\operatorname{Conv}(\mathbb{R}^{n})$.*

Proof.

By Lemma 3, epi-convergence coincides with pointwise convergence on finite functions in $\operatorname{Conv}(\mathbb{R}^{n})$. Therefore, it is easy to see that $\operatorname{Conv_{p.a.}}(\mathbb{R}^{n})$ is epi-dense in the finite elements of $\operatorname{Conv}(\mathbb{R}^{n})$. Now for arbitrary $u\in\operatorname{Conv}(\mathbb{R}^{n})$ it follows from Lemma 9 that $e_{t}u$ is a finite element of $\operatorname{Conv}(\mathbb{R}^{n})$. Since Lemma 10 gives that $\operatorname{epi-lim}_{t\to 0^{+}}e_{t}u=u$, the finite elements of $\operatorname{Conv}(\mathbb{R}^{n})$ are a dense subset of $\operatorname{Conv}(\mathbb{R}^{n})$. Since denseness is transitive, the piecewise affine functions are a dense subset of $\operatorname{Conv}(\mathbb{R}^{n})$. ∎

2 Valuations on Convex Functions

The functionals that appear in the theorem are discussed. It is shown that they are continuous, $\operatorname{SL}(n)$ and translation invariant valuations on $\operatorname{Conv}(\mathbb{R}^{n})$.

Lemma 12**.**

For $\zeta\in C(\mathbb{R})$, the map

[TABLE]

is a continuous, $\operatorname{SL}(n)$ and translation invariant valuation on $\operatorname{Conv}(\mathbb{R}^{n})$.

Proof.

Let $u\in\operatorname{Conv}(\mathbb{R}^{n})$. Since

[TABLE]

for every $\phi\in\operatorname{SL}(n)$ and translation $\tau:\mathbb{R}^{n}\to\mathbb{R}^{n}$, (11) defines an $\operatorname{SL}(n)$ and translation invariant map. If $u,v\in\operatorname{Conv}(\mathbb{R}^{n})$ are such that $u\wedge v\in\operatorname{Conv}(\mathbb{R}^{n})$, then clearly

[TABLE]

By [8, Lemma 3.7] we have

[TABLE]

Hence, a function $\zeta\in C(\mathbb{R})$ composed with the minimum of a function $u\in\operatorname{Conv}(\mathbb{R}^{n})$ defines a valuation on $\operatorname{Conv}(\mathbb{R}^{n})$. The continuity of (11) follows from Lemma 5. ∎

Let $\zeta\in C(\mathbb{R})$ be non-negative. For $u\in\operatorname{Conv}(\mathbb{R}^{n})$, define

[TABLE]

We want to investigate conditions on $\zeta$ such that $\operatorname{Z}_{\zeta}$ defines a continuous valuation on $\operatorname{Conv}(\mathbb{R}^{n})$.

It is easy to see, that in order for $\operatorname{Z}_{\zeta}(u)$ to be finite for every $u\in\operatorname{Conv}(\mathbb{R}^{n})$, it is necessary for $\zeta$ to have finite $(n-1)$-st moment. Indeed, if $u(x)=|x|$, then

[TABLE]

where $v_{n}$ is the volume of the $n$-dimensional unit ball. We will see in Lemma 14, that this condition is also sufficient. For this, we require the following result.

Lemma 13**.**

Let $u_{k}$ be a sequence in $\operatorname{Conv}(\mathbb{R}^{n})$ with epi-limit $u\in\operatorname{Conv}(\mathbb{R}^{n})$. If $\zeta\in C(\mathbb{R})$ is non-negative with finite $(n-1)$-st moment, then, for every $\varepsilon>0$, there exist $t_{0}\in\mathbb{R}$ and $k_{0}\in\mathbb{N}$ such that

[TABLE]

for every $t\geq t_{0}$ and $k\geq k_{0}$.

Proof.

Without loss of generality, let $\min_{x\in\mathbb{R}^{n}}u(x)=u(0)$. By the definition of epi-convergence, there exists a sequence $x_{k}$ in $\mathbb{R}^{n}$ such that $x_{k}\to 0$ and $u_{k}(x_{k})\to u(0)$. Therefore, there exists $k_{0}\in\mathbb{N}$ such that $|x_{k}|<1$ and $u_{k}(x_{k})<u(0)+1$ for every $k\geq k_{0}$. By Lemma 8, there exist constants $a>0$ and $\bar{b}\in\mathbb{R}$, such that

[TABLE]

for every $x\in\mathbb{R}^{n}$ and $k\in\mathbb{N}$. Setting $\widetilde{u}_{k}(x)=u_{k}(x-x_{k})$, we have

[TABLE]

for every $k\geq k_{0}$. Hence, with $b=\bar{b}-a$, we have

[TABLE]

for every $x\in\mathbb{R}^{n}$ and $k\geq k_{0}$.

We write $x=r\omega$ with $r\in[0,+\infty)$ and $\omega\in{\mathbb{S}}^{n-1}$. For $u(r\omega)\geq 1$, we obtain from (13) that

[TABLE]

for every $r\in[0,+\infty)$, $\omega\in{\mathbb{S}}^{n-1}$ and $k\geq k_{0}$, where $c$ only depends on $a,b$ and the dimension $n$. Now choose $\bar{t}_{0}\geq\max\{1,2(u(0)+1)-b\}$. Then for all $t\geq\bar{t}_{0}$

[TABLE]

For $\omega\in{\mathbb{S}}^{n-1}$, let $v_{\omega}(r)=u(r\omega)$. The function $v_{\omega}$ is non-decreasing and convex on $[0,+\infty)$. So, in particular, the left and right derivatives, $v^{\prime}_{\omega,l},v^{\prime}_{\omega,r}$ of $v_{\omega}$ exist and for the subgradient $\partial v_{\omega}(r)=[v^{\prime}_{\omega,l},v^{\prime}_{\omega,r}]$, it follows from $r<\bar{r}$ that $\eta\leq\bar{\eta}$ for $\eta\in\partial v_{\omega}(r)$ and $\bar{\eta}\in\partial v_{\omega}(\bar{r})$.

For $t\geq\bar{t}_{0}$, set

[TABLE]

For every $\omega\in{\mathbb{S}}^{n-1}$, the set $D_{\omega}(t)$ is either empty or there exists

[TABLE]

and $v_{\omega}(r_{\omega}(t))=t$. Therefore, if $D_{\omega}(t)$ is non-empty, we have

[TABLE]

for $\xi\in\partial v_{\omega}(r_{\omega}(t))$. Hence, it follows from (16) and (15) that

[TABLE]

for all $r\in D_{\omega}(t)$, $\vartheta\in\partial v_{\omega}(r)$ and $\xi\in\partial v_{\omega}(r_{\omega}(t))$. Similarly, setting $\widetilde{v}_{k,\omega}(r)=\widetilde{u}_{k}(r\omega)$ and

[TABLE]

it is easy to see that $\widetilde{v}_{k,\omega}$ is convex on $[0,+\infty)$ and monotone increasing on $\widetilde{D}_{k,\omega}(t)$ for all $k\geq k_{0}$. By the choice of $\bar{t}_{0}$ and (15), for $t\geq\bar{t}_{0}$

[TABLE]

for all $r\in D_{k,\omega}(t)$, $k\geq k_{0}$ and $\vartheta\in\partial\widetilde{v}_{k,\omega}(r)$. Recall, that as a convex function $v_{\omega}$ is locally Lipschitz and differentiable almost everywhere on the interior of its domain. Using polar coordinates, (14) and the substitution $v_{\omega}(r)=s$, we obtain from (17) that

[TABLE]

for every $t\geq\bar{t}_{0}$. In the same way,

[TABLE]

for every $t\geq\bar{t}_{0}$ and $k\geq k_{0}$ with the same constant ${c}$ as in (18). The statement now follows, since $\zeta$ is non-negative and has finite $(n-1)$-st moment. ∎

Lemma 14**.**

Let $\zeta\in C(\mathbb{R})$ be non-negative. Then $\operatorname{Z}_{\zeta}(u)<+\infty$ for every $u\in\operatorname{Conv}(\mathbb{R}^{n})$ if and only if $\zeta$ has finite $(n-1)$-st moment.

Proof.

As already pointed out in (12), it is necessary for $\zeta$ to have finite $(n-1)$-st moment in order for $\operatorname{Z}_{\zeta}$ to be finite.

Now let $u\in\operatorname{Conv}(\mathbb{R}^{n})$ be arbitrary, let $\zeta$ have finite $(n-1)$-st moment and let $u_{\min}=\min_{x\in\mathbb{R}^{n}}u(x)$. By Lemma 13, there exists $t\in\mathbb{R}$ such that

[TABLE]

It follows that

[TABLE]

and hence $\operatorname{Z}_{\zeta}(u)<\infty$. ∎

Lemma 15**.**

For $\zeta\in C(\mathbb{R})$ non-negative and with finite $(n-1)$-st moment, the functional $\operatorname{Z}_{\zeta}$ is continuous on $\operatorname{Conv}(\mathbb{R}^{n})$.

Proof.

Let $u\in\operatorname{Conv}(\mathbb{R}^{n})$ and let $u_{k}$ be a sequence in $\operatorname{Conv}(\mathbb{R}^{n})$ such that $u_{k}\stackrel{{\scriptstyle epi}}{{\longrightarrow}}u$. Set $u_{\min}=\min_{x\in\mathbb{R}^{n}}u(x)$. By Lemma 13, it is enough to show that

[TABLE]

for every fixed $t>u_{\min}$. Lemma 5 implies that $\{u_{k}\leq t\}\to\{u\leq t\}$ in the Hausdorff metric. By Lemma 8, there exists $b\in\mathbb{R}$ such that $u(x),u_{k}(x)>b$ for $x\in\mathbb{R}^{n}$ and $k\in\mathbb{N}$. Set $c=\max_{s\in[b,t]}\zeta(s)\geq 0$. We distinguish the following cases.

First, let $\dim(\operatorname{dom}u)<n$. In this case $V_{n}(\{u\leq t\})=0$ and since volume is continuous on convex sets, $V_{n}(\{u_{k}\leq t\})\to 0$. Hence,

[TABLE]

Second, let $\dim(\operatorname{dom}u)=n$. In this case, $\{u\leq t\}$ is a set in ${\mathcal{K}}^{n}$ with non-empty interior. Therefore, for $\varepsilon>0$ there exist $k_{0}\in\mathbb{N}$ and $C\in{\mathcal{K}}^{n}$ such that for every $k\geq k_{0}$ the following hold:

[TABLE]

[TABLE]

[TABLE]

where $C^{c}$ is the complement of $C$. Note, that $u(x),u_{k}(x)\in[b,t]$ for $x\in C$ and $k\geq k_{0}$. Since $C\subset\operatorname{int}\operatorname{dom}u$, Lemma 2 implies that $u_{k}$ converges to $u$ uniformly on $C$. Since $\zeta$ is continuous, the restriction of $\zeta$ to $[b,t]$ is uniformly continuous. Hence, $\zeta\circ u_{k}$ converges uniformly to $\zeta\circ u$ on $C$. Therefore, there exists $k_{1}\geq k_{0}$ such that

[TABLE]

for all $x\in C$ and $k\geq k_{1}$. This gives

[TABLE]

for $k\geq k_{1}$. The statement now follows, since $\varepsilon>0$ was arbitrary. ∎

Lemma 16**.**

For $\zeta\in C(\mathbb{R})$ non-negative and with finite $(n-1)$-st moment, the functional $\operatorname{Z}_{\zeta}$ is an $\operatorname{SL}(n)$ and translation invariant valuation on $\operatorname{Conv}(\mathbb{R}^{n})$.

Proof.

It is easy to see that $\operatorname{Z}_{\zeta}$ is $\operatorname{SL}(n)$ and translation invariant. It remains to show the valuation property. Let $u,v\in\operatorname{Conv}(\mathbb{R}^{n})$ be such that $u\wedge v\in\operatorname{Conv}(\mathbb{R}^{n})$. We have

[TABLE]

[TABLE]

Hence,

[TABLE]

and the valuation property is proved. ∎

3 Valuations on Cone and Indicator Functions

Let ${\mathcal{K}}^{n}_{0}$ be the set of compact convex sets which contain the origin. For $K\in{\mathcal{K}}^{n}_{0}$, we define the convex function $\ell_{K}:\mathbb{R}^{n}\to[0,\infty]$ via

[TABLE]

where $\operatorname{pos}$ denotes the positive hull. This means that the epigraph of $\ell_{K}$ is a cone with apex at the origin and $\{\ell_{K}\leq t\}=t\,K$ for all $t\geq 0$. It is easy to see that $\ell_{K}$ is an element of $\operatorname{Conv}(\mathbb{R}^{n})$ for all $K\in{\mathcal{K}}^{n}_{0}$. We have $\operatorname{dom}\ell_{K}=\mathbb{R}^{n}$ if and only if $K$ contains the origin in its interior. If $P\in{\mathcal{P}}^{n}_{0}$ contains the origin in its interior, then $\ell_{P}\in\operatorname{Conv_{p.a.}}(\mathbb{R}^{n})$. For $K\in{\mathcal{K}}^{n}_{0}$ and $t\in\mathbb{R}$, we call the function $\ell_{K}+t$ a cone function and we call the function $\mathrm{I}_{K}+t$ an indicator function. Cone and indicator functions play a special role in our proof.

The next result shows that to classify continuous and translation invariant valuations on $\operatorname{Conv}(\mathbb{R}^{n})$, it is enough to know the behavior of these valuations on cone functions. The main argument of the following lemma is due to [23, Lemma 8], where it was used for functions on Sobolev spaces.

Lemma 17**.**

Let $\langle A,+\rangle$ be a topological abelian semigroup with cancellation law and let $\,\operatorname{Z}_{1},\operatorname{Z}_{2}:\operatorname{Conv}(\mathbb{R}^{n})\to\langle A,+\rangle$ be continuous, translation invariant valuations. If $\,\operatorname{Z}_{1}(\ell_{P}+t)=\operatorname{Z}_{2}(\ell_{P}+t)$ for every $P\in{\mathcal{P}}_{0}^{n}$ and $t\in\mathbb{R}$, then $\operatorname{Z}_{1}\equiv\operatorname{Z}_{2}$ on $\operatorname{Conv}(\mathbb{R}^{n})$.

Proof.

By Lemma 11 and the continuity of $\operatorname{Z}_{1}$ and $\operatorname{Z}_{2}$, it suffices to show that $\operatorname{Z}_{1}$ and $\operatorname{Z}_{2}$ coincide on $\operatorname{Conv_{p.a.}}(\mathbb{R}^{n})$. So let $u\in\operatorname{Conv_{p.a.}}(\mathbb{R}^{n})$ and set $U=\operatorname{epi}u$. Note, that $U$ is a convex polyhedron in $\mathbb{R}^{n+1}$ and that none of the facet hyperplanes of $U$ is parallel to the $x_{n+1}$-axis. Here, we say that a hyperplane $H$ in $\mathbb{R}^{n+1}$ is a facet hyperplane of $U$ if its intersection with the boundary of $U$ has positive $n$-dimensional Hausdorff measure. Furthermore, we call $U$ singular if $U$ has $n$ facet hyperplanes whose intersection contains a line parallel to $\{x_{n+1}=0\}$. Since $\operatorname{Z}_{1}$ and $\operatorname{Z}_{2}$ are continuous, we can assume that $U$ is not singular.

Since $U$ is not singular and $u\in\operatorname{Conv_{p.a.}}(\mathbb{R}^{n})$, there exists a unique vertex, $\bar{p}$ of $U$ with smallest $x_{n+1}$ coordinate. We use induction on the number $m$ of facet hyperplanes of $U$ that are not passing through $\bar{p}$. If $m=0$, then there exist $P\in{\mathcal{P}}_{0}^{n}$ and $t\in\mathbb{R}$ such that $u$ is a translate of $\ell_{P}+t$. Since $\operatorname{Z}_{1}$ and $\operatorname{Z}_{2}$ are translation invariant, it follows that $\operatorname{Z}_{1}(u)=\operatorname{Z}_{2}(u)$.

Now let $U$ have $m>0$ facet hyperplanes that are not passing through $\bar{p}$ and assume that $\operatorname{Z}_{1}$ and $\operatorname{Z}_{2}$ coincide for all functions with at most $(m-1)$ such facet hyperplanes. Let $p_{0}=(x_{0},u(x_{0}))\in\mathbb{R}^{n+1}$ where $x_{0}\in\mathbb{R}^{n}$ is a vertex of $U$ with maximal $x_{n+1}$-coordinate and let $H_{1},\ldots,H_{j}$ be the facet hyperplanes of $U$ through $p_{0}$ such that the corresponding facets of $U$ have infinite $n$-dimensional volume. Note, that $H_{1},\ldots,H_{j}$ do not contain $\bar{p}$ and therefore there is at least one such hyperplane. Define $\bar{U}$ as the polyhedron bounded by the intersection of all facet hyperplanes of $U$ with the exception of $H_{1},\ldots,H_{j}$. Since $U$ is not singular, there exists a function $\bar{u}\in\operatorname{Conv_{p.a.}}(\mathbb{R}^{n})$ with $\operatorname{dom}\bar{u}=\mathbb{R}^{n}$ such that $\bar{U}=\operatorname{epi}\bar{u}$. Note, that $\bar{U}$ has at most $(m-1)$ facet hyperplanes not containing $\bar{p}$. Hence, by the induction hypothesis

[TABLE]

Let $\overline{H}_{1},\ldots,\overline{H}_{i}$ be the facet hyperplanes of $\bar{U}$ that contain $p_{0}$ such that the corresponding facets of $\bar{U}$ have infinite $n$-dimensional volume. Choose suitable hyperplanes $\overline{H}_{i+1},\ldots,\overline{H}_{k}$ not parallel to the $x_{n+1}$-axis and containing $p_{0}$ so that the hyperplanes $\overline{H}_{1},\ldots,\overline{H}_{k}$ bound a polyhedral cone with apex $p_{0}$ that is contained in $\bar{U}$, has $\overline{H}_{1},\ldots,\overline{H}_{i}$ among its facet hyperplanes and contains $\{x_{0}\}\times[u(x_{0}),+\infty)$. Define $\ell$ as the piecewise affine function determined by this polyhedral cone. Notice, that $\ell$ is a translate of $\ell_{P}+u(x_{0})$, where $P\in{\mathcal{P}}_{0}^{n}$ is the projection onto the first $n$ coordinates of the intersection of the polyhedral cone with $\{x_{n+1}=u(x_{0})+1\}$. Hence, $\operatorname{Z}_{1}$ and $\operatorname{Z}_{2}$ coincide on $\ell$. Set $\bar{\ell}=u\vee\ell$. The epigraph of $\bar{\ell}$ is again a polyhedral cone with apex $p_{0}$. Hence $\bar{\ell}$ is a translate of $\ell_{\bar{P}}+u(x_{0})$ with $\bar{P}\in{\mathcal{P}}_{0}^{n}$ since it is bounded by hyperplanes containing $p_{0}$ that are not parallel to the $x_{n+1}$-axis. Therefore, $\operatorname{Z}_{1}$ and $\operatorname{Z}_{2}$ also coincide on $\bar{\ell}$. We now have

[TABLE]

From the valuation property of $\operatorname{Z}_{i}$, $i=1,2$, we obtain

[TABLE]

which completes the proof. ∎

Next, we study the behavior of a continuous and $\operatorname{SL}(n)$ invariant valuation on cone and indicator functions.

Lemma 18**.**

If $\,\operatorname{Z}:\operatorname{Conv}(\mathbb{R}^{n})\to\mathbb{R}$ is a continuous and $\operatorname{SL}(n)$​ invariant valuation, then there exist continuous functions $\psi_{0},\psi_{n},\zeta_{0},\zeta_{n}:\mathbb{R}\to\mathbb{R}$ such that

[TABLE]

for every $P\in{\mathcal{P}}_{0}^{n}$ and $t\in\mathbb{R}$.

Proof.

For $t\in\mathbb{R}$, define $\operatorname{Z}_{t}:{\mathcal{P}}_{0}^{n}\to\mathbb{R}$ as

[TABLE]

It is easy to see that $\operatorname{Z}_{t}$ defines a continuous, $\operatorname{SL}(n)$ invariant valuation on ${\mathcal{P}}_{0}^{n}$ for every $t\in\mathbb{R}$. Therefore, by (4), for every $t\in\mathbb{R}$ there exist constants $c_{0,t},c_{n,t}\in\mathbb{R}$ such that

[TABLE]

for every $P\in{\mathcal{P}}_{0}^{n}$. This defines two functions $\psi_{0}(t)=c_{0,t}$ and $\psi_{n}(t)=c_{n,t}$. Taking $P\in{\mathcal{P}}_{0}^{n}$ with $\dim P<n$, we have $V_{n}(P)=0$. By the continuity of $\operatorname{Z}$,

[TABLE]

is continuous, which implies that $\psi_{0}$ is a continuous function. Similarly, taking $Q\in{\mathcal{P}}_{0}^{n}$ with $V_{n}(Q)>0$, we see that

[TABLE]

can be expressed as the difference of two continuous functions and is therefore continuous itself. Using $P\mapsto\operatorname{Z}(\mathrm{I}_{P}+t)$ we get the corresponding results for the functions $\zeta_{0}$ and $\zeta_{n}$. ∎

For a continuous and $\operatorname{SL}(n)$ invariant valuation $\operatorname{Z}:\operatorname{Conv}(\mathbb{R}^{n})\to\mathbb{R}$, we call the functions $\psi_{0}$ and $\psi_{n}$ from Lemma 18 the cone growth functions of $\operatorname{Z}$. The functions $\zeta_{0}$ and $\zeta_{n}$ are its indicator growth functions. By Lemma 17, we immediately get the following result.

Lemma 19**.**

Every continuous, $\operatorname{SL}(n)$ and translation invariant valuation $\operatorname{Z}:\operatorname{Conv}(\mathbb{R}^{n})\to\mathbb{R}$ is uniquely determined by its cone growth functions.

In order to classify valuations, we want to determine how the cone growth functions and the indicator growth functions are related.

Lemma 20**.**

For $k\geq 1$, let $\operatorname{Z}:\operatorname{Conv}(\mathbb{R}^{k})\to\mathbb{R}$ be a continuous, translation invariant valuation and let $\psi\in C(\mathbb{R})$. If

[TABLE]

for every $P\in{\mathcal{P}}_{0}^{k}$ and $t\in\mathbb{R}$, then

[TABLE]

for every $t\in\mathbb{R}$. In particular, $\psi$ is $k$-times differentiable.

Proof.

To explain the idea of the proof, we first consider the case $k=1$. For $h>0$, let $u_{h}=\ell_{[0,1/h]}$, that is, $u^{h}(x)=+\infty$ for $x<0$ and $u^{h}(x)=h\,x$ for $x\geq 0$. Define $v^{h}:\mathbb{R}\to[0,+\infty]$ by $v^{h}=u^{h}+\mathrm{I}_{[0,1]}$. Since $\operatorname{Z}$ is a translation invariant valuation and by (19), we obtain

[TABLE]

for $t\in\mathbb{R}$. As $h\to 0$, the epi-limit of $v^{h}+t$ is $\mathrm{I}_{[0,1]}+t$. Since $\operatorname{Z}$ is continuous, we thus obtain

[TABLE]

for $t\in\mathbb{R}$. Hence $\psi$ is differentiable from the right at every $t\in\mathbb{R}$. Since $v^{h}+t-h\stackrel{{\scriptstyle epi}}{{\longrightarrow}}\mathrm{I}_{[0,1]}+t$ as $h\to 0$, we also obtain

[TABLE]

Hence $\psi$ is also differentiable from the left at every $t\in\mathbb{R}$ and $\operatorname{Z}(\mathrm{I}_{[0,1]}+t)=-\psi^{\prime}(t)$. This concludes the proof for $k=1$.

Next, let $\{e_{1},\ldots,e_{k}\}$ denote the standard basis of $\mathbb{R}^{k}$ and set $e_{0}=0$. For $h=(h_{1},\ldots,h_{k})$ with $0<h_{1}\leq\cdots\leq h_{k}$ and $0\leq i<k$, define the function $u_{i}^{h}$ through its sublevel sets as

[TABLE]

for every $s\geq 0$. Let $u_{k}^{h}=\mathrm{I}_{[0,1]^{k}}$. Note, that $u_{i}^{h}$ does not depend on $h_{j}$ for $0\leq j\leq i$. We use induction on $i$ to show that $u_{i}^{h}\in\operatorname{Conv}(\mathbb{R}^{k})$ and that

[TABLE]

for every $t\in\mathbb{R}$ and $0\leq i\leq k$, where $\psi^{(i)}(t)=\tfrac{\,\mathrm{d}^{i}}{\,\mathrm{d}t^{i}}\psi(t)$.

For $i=0$, set $P_{h}=\operatorname{conv}\{0,e_{1}/h_{1},\ldots,e_{k}/h_{k}\}\in{\mathcal{P}}_{0}^{k}$ and note that $u_{0}^{h}=\ell_{P_{h}}\in\operatorname{Conv}(\mathbb{R}^{k})$. Hence, by the assumption on $\operatorname{Z}$, we have

[TABLE]

Now assume that the statement holds true for $i\geq 0$. Define the function $v_{i+1}^{h}$ by

[TABLE]

for every $s\in\mathbb{R}$. Since $\operatorname{epi}v_{i+1}^{h}=\operatorname{epi}u_{i}^{h}\cap\{x_{i+1}\leq 1\}$, it is easy to see that $v_{i+1}^{h}\in\operatorname{Conv}(\mathbb{R}^{k})$. As $h_{i+1}\to 0$, we have epi-convergence of $v_{i+1}^{h}$ to $u_{i+1}^{h}$. Lemma 2 implies that $u_{i+1}^{h}$ is a convex function and hence $u_{i+1}^{h}\in\operatorname{Conv}(\mathbb{R}^{k})$. Now, let $\tau_{i+1}$ be the translation $x\mapsto x+e_{i+1}$. Note that

[TABLE]

[TABLE]

for every $s\in\mathbb{R}$. Since $\operatorname{Z}$ is a continuous, translation invariant valuation and $\operatorname{Z}(\ell_{P}+t)=0$ for $P\in{\mathcal{P}}_{0}^{k}$ with $\dim(P)<k$, Lemma 17 and its proof imply that $\operatorname{Z}$ vanishes on all functions $u\in\operatorname{Conv}(\mathbb{R}^{k})$ with $\operatorname{dom}u\subset H$, where $H$ is a hyperplane in $\mathbb{R}^{k}$. Hence,

[TABLE]

Thus, by the valuation property

[TABLE]

Using the induction assumption and the translation invariance of $\operatorname{Z}$, we obtain

[TABLE]

As $h_{i+1}\to 0$, the continuity of $\operatorname{Z}$ shows that

[TABLE]

Hence $\psi^{(i)}$ is differentiable from the right. Similarly, we have $v_{i+1}^{h}+t-h_{i+1}\stackrel{{\scriptstyle epi}}{{\longrightarrow}}u_{i+1}^{h}$ as $h_{i+1}\to 0$ and thus

[TABLE]

which shows that $\psi^{(i)}$ is differentiable from the left and therefore,

[TABLE]

for every $t\in\mathbb{R}$. ∎

Lemma 21**.**

If $\,\operatorname{Z}:\operatorname{Conv}(\mathbb{R}^{n})\to\mathbb{R}$ is a continuous, $\operatorname{SL}(n)\!$ and translation invariant valuation, then the growth functions $\psi_{0}$ and $\zeta_{0}$ coincide and

[TABLE]

for every $t\in\mathbb{R}$.

Proof.

Since $\ell_{\{0\}}=\mathrm{I}_{\{0\}}$, Lemma 18 implies that

[TABLE]

for every $t\in\mathbb{R}$.

Now define $\bar{\operatorname{Z}}:\operatorname{Conv}(\mathbb{R}^{n})\to\mathbb{R}$ as

[TABLE]

By Lemma 12, the functional $\bar{\operatorname{Z}}$ is a continuous, $\operatorname{SL}(n)$ and translation invariant valuation that satisfies

[TABLE]

and

[TABLE]

for every $P\in{\mathcal{P}}_{0}^{n}$ and $t\in\mathbb{R}$. Hence, by Lemma 20,

[TABLE]

for every $t\in\mathbb{R}$. ∎

Lemma 22**.**

If $\,\operatorname{Z}:\operatorname{Conv}(\mathbb{R}^{n})\to\mathbb{R}$ is a continuous, $\operatorname{SL}(n)$ and translation invariant valuation, then its cone growth function $\psi_{n}$ satisfies

[TABLE]

Proof.

Let $\{e_{1},e_{2},\ldots,e_{n}\}$ be the standard basis of $\mathbb{R}^{n}$ and let

[TABLE]

For $s>0$, define $u_{s}\in\operatorname{Conv}(\mathbb{R}^{n})$ by its epigraph as $\operatorname{epi}u_{s}=\operatorname{epi}\ell_{P}\cap\{x_{1}\leq\tfrac{s}{2}\}$. Note, that for $t\geq 0$ this gives $\{u_{s}\leq t\}=tP\cap\{x_{1}\leq\tfrac{s}{2}\}$. Let $\tau_{s}$ be the translation $x\mapsto x+s\,\tfrac{e_{1}+e_{2}}{2}$ and define $\ell_{P,s}(x)=\ell_{P}(x)\circ\tau_{s}^{-1}+s$ and similarly $\ell_{Q,s}(x)=\ell_{Q}(x)\circ\tau_{s}^{-1}+s$. We will now show that

[TABLE]

or equivalently

[TABLE]

which is the same as

[TABLE]

for every $t\in\mathbb{R}$. Indeed, it is easy to see, that (20) holds for all $t<s$. Therefore, fix an arbitrary $t\geq s$. We have

[TABLE]

This can be rewritten as

[TABLE]

Hence

[TABLE]

and

[TABLE]

From the valuation property of $\operatorname{Z}$ we now get

[TABLE]

By Lemma 18 and since $V_{n}(Q)=0$, we have

[TABLE]

As $s\to\infty$, we obtain $u_{s}\stackrel{{\scriptstyle epi}}{{\longrightarrow}}\ell_{P}$ and therefore

[TABLE]

Since $V_{n}(P)>0$, this shows that $\psi_{n}(s)\to 0$. ∎

Lemma 21 shows that for a continuous, $\operatorname{SL}(n)$ and translation invariant valuation $\operatorname{Z}$ the indicator growth functions $\zeta_{0}$ and $\zeta_{n}$ coincide with its cone growth function $\psi_{0}$ and up to a constant factor with the $n$-th derivative of its cone growth function $\psi_{n}$, respectively. Since Lemma 22 shows that $\lim_{t\to\infty}\psi_{n}(t)=0$, the cone growth functions $\psi_{0}$ and $\psi_{n}$ are completely determined by the indicator growth functions of $\operatorname{Z}$. Hence Lemma 19 immediately implies the following result.

Lemma 23**.**

Every continuous, $\operatorname{SL}(n)$ and translation invariant valuation $\operatorname{Z}:\operatorname{Conv}(\mathbb{R}^{n})\to\mathbb{R}$ is uniquely determined by its indicator growth functions.

We also require the following result.

Lemma 24**.**

Let $\zeta\in C(\mathbb{R})$ have constant sign on $[t_{0},\infty)$ for some $t_{0}\in\mathbb{R}$. If there exist $n\in\mathbb{N}$, $c_{n}\in\mathbb{R}$ and $\psi\in C^{n}(\mathbb{R})$ with $\lim_{t\to+\infty}\psi(t)=0$ such that

[TABLE]

for $t\geq t_{0}$, then

[TABLE]

Proof.

Since we can always consider $\widetilde{\psi}(t)=\pm c_{n}\,\psi(t)$ instead of $\psi(t)$, we assume that $c_{n}=1$ and $\zeta\geq 0$. To prove the statement, we use induction on $n$ and start with the case $n=1$. For $t_{1}>t_{0}$,

[TABLE]

Hence, it follows from the assumption for $\psi$ that

[TABLE]

This proves the statement for $n=1$.

Let $n\geq 2$ and assume that the statement holds true for $n-1$. Since $\zeta\geq 0$, the function $\psi^{(n-1)}$ is increasing. Therefore, the limit

[TABLE]

exists. Moreover, $\psi^{(n-1)}$ has constant sign on $[\bar{t}_{0},+\infty)$ for some $\bar{t}_{0}\geq t_{0}$. By the induction hypothesis,

[TABLE]

which is only possible if $c=0$. In particular, $\psi^{(n-1)}(t)\leq 0$ for all $t\geq\bar{t}_{0}$.

Using integration by parts, we obtain

[TABLE]

Since $t^{n-1}\psi^{(n)}(t)\geq 0$ for $t\geq\max\{0,t_{0}\}$, we have

[TABLE]

Hence, (21) implies that $t_{1}^{n-1}\psi^{(n-1)}(t_{1})$ converges to

[TABLE]

Since $t_{1}^{n-1}\psi^{(n-1)}(t_{1})\leq 0$ for $t_{1}\geq\max\{\bar{t}_{0},0\}$, we conclude that $d$ is not $+\infty$. ∎

4 Proof of the Theorem

If $\zeta_{0}:\mathbb{R}\to[0,\infty)$ is continuous and $\zeta_{n}:\mathbb{R}\to[0,\infty)$ is continuous with finite $(n-1)$-st moment, then Lemmas 12 and 16 show that

[TABLE]

defines a non-negative, continuous, $\operatorname{SL}(n)$ and translation invariant valuation on $\operatorname{Conv}(\mathbb{R}^{n})$.

Conversely, let $\operatorname{Z}:\operatorname{Conv}(\mathbb{R}^{n})\to[0,\infty)$ be a continuous, $\operatorname{SL}(n)$ and translation invariant valuation on $\operatorname{Conv}(\mathbb{R}^{n})$ with indicator growth functions $\zeta_{0}$ and $\zeta_{n}$. For a polytope $P\in{\mathcal{P}}_{0}^{n}$ with $\dim P<n$, Lemma 18 implies that

[TABLE]

for every $t\in\mathbb{R}$. Hence, $\zeta_{0}$ is a non-negative and continuous function. Similarly, for $Q\in{\mathcal{P}}_{0}^{n}$ with $V_{n}(Q)>0$, we have

[TABLE]

for every $t\in\mathbb{R}$ and $s>0$. Therefore, also $\zeta_{n}$ is a non-negative and continuous function. By Lemmas 21, 22 and 24, the growth function $\zeta_{n}$ has finite $(n-1)$-st moment. Finally, for $u=\mathrm{I}_{P}+t$ with $P\in{\mathcal{P}}_{0}^{n}$ and $t\in\mathbb{R}$, we obtain that

[TABLE]

By the first part of the proof,

[TABLE]

defines a non-negative, continuous, $\operatorname{SL}(n)$ and translation invariant valuation on $\operatorname{Conv}(\mathbb{R}^{n})$. Thus Lemma 23 completes the proof of the theorem.

Acknowledgments

The work of Monika Ludwig and Fabian Mussnig was supported, in part, by Austrian Science Fund (FWF) Project P25515-N25. The work of Andrea Colesanti was supported by the G.N.A.M.P.A. and by the F.I.R. project 2013: Geometrical and Qualitative Aspects of PDE’s.