1 Introduction↩︎

In his celebrated work, Hadwiger [1] classified continuous, translation and rotation invariant, scalar-valued valuations on convex bodies in \(\mathbb{R}^n\) and obtained a characterization of intrinsic volumes. Schneider [2] extended Hadwiger’s results to measure-valued valuations. He established a classification of weak\(^*\) continuous, locally determined, translation invariant, and rotation equivariant valuations on convex bodies in \(\mathbb{R}^n\) with values in the space of signed Radon measures on the unit sphere and a characterization of area measures (see [3] for comprehensive information on intrinsic volumes, area measures, and valuations on convex bodies).

Dual intrinsic volumes were introduced by Lutwak [4] within the framework of the dual Brunn–Minkowski theory. They are defined on the space \(\mathcal{S}_{\scriptscriptstyle 0}^n\) of star bodies in \(\mathbb{R}^n\), equipped with the topology induced by the radial distance (see Section 2 for precise definitions). A map \(\operatorname{Z}\) defined on \(\mathcal{S}_{\scriptscriptstyle 0}^n\) with values in an additive abelian semigroup is a valuation if \[\label{eq:val} \operatorname{Z}(K\cup L)+\operatorname{Z}(K\cap L)=\operatorname{Z}(K)+ \operatorname{Z}(L).\tag{1}\] for every \(K, L\in\mathcal{S}_{\scriptscriptstyle 0}^n\). In [5]–[7], extending results by Klain [8], [9], a classification of continuous, scalar-valued valuations on \(\mathcal{S}_{\scriptscriptstyle 0}^n\) was established, and a characterization of dual intrinsic volumes was obtained (see [3], [10] for more information on the dual Brunn–Minkowski theory and its applications).

The aim of this paper is to establish classification results for measure-valued valuations on \(\mathcal{S}_{\scriptscriptstyle 0}^n\). Let \(M({S^{n-1}})_{\scriptscriptstyle +}\) denote the space of non-negative Radon measures on the unit sphere \({S^{n-1}}\) and \(\mathbb{R}_{\scriptscriptstyle +}\) the non-negative real numbers. Our main classification result is given in the following theorem.

Here, \(\rho_L\) is the radial function of \(L\) (see Subsection 2.1 for the definition). We say that a map \(U\colon \mathbb{R}_{\scriptscriptstyle +}\times {S^{n-1}}\to M({S^{n-1}})_{\scriptscriptstyle +}\) is \(\mu\)-strong Carathéodory if the map \((\lambda, s)\mapsto \langle U(\lambda, s),g\rangle\) is strong Carathéodory (see Subsection 2.4) with respect to \(\mu\) for every \(g\in C({S^{n-1}})\) (with the pairing defined in 5 ). We call the map \(U\) uniformly controlled if there is a function \(\zeta\colon \mathbb{R}_{\scriptscriptstyle +}\to \mathbb{R}_{\scriptscriptstyle +}\) such that \[\vert \langle U(\lambda, s), g\rangle\vert \leq \|g\|_{\scriptscriptstyle \infty}\, \zeta(\lambda)\] for every \(\lambda\in\mathbb{R}_{\scriptscriptstyle +}\), every \(s\in{S^{n-1}}\), and every \(g\in C({S^{n-1}})\).

Valuations that intertwine rotations are particularly interesting. We call a valuation \(\operatorname{Z}\colon \mathcal{S}_{\scriptscriptstyle 0}^n\to M({S^{n-1}})_{\scriptscriptstyle +}\) rotation equivariant if \[\langle \operatorname{Z}(\phi L),\phi g\rangle = \langle \operatorname{Z}(L),g\rangle\] for all \(L\in\mathcal{S}_{\scriptscriptstyle 0}^n\), \(g\in C({S^{n-1}})\), and \(\phi\in\operatorname{O}(n)\), the orthogonal group in \(\mathbb{R}^n\).

As a consequence of Theorem 1, we will prove the following classification of rotation equivariant valuations. Let \(\mathcal{H}^k\) denote \(k\)-dimensional Hausdorff measure.

Here, \(\operatorname{\overline{R}}g(s,\alpha)\) is the average of \(g\) in the set \(\{r\in{S^{n-1}}:\langle r,s\rangle=\alpha\}\) (see Subsection 6.1) and we also write \(\tau_\lambda\) for the measure \(\tau(\lambda)\).

Huang, Lutwak, Yang, and Zhang [11] introduced the \(q\)-th dual area measure \(\tilde{S}_q(L)\) for \(L\in\mathcal{S}_{\scriptscriptstyle 0}^n\) and \(q\in\mathbb{R}\) within the dual Brunn–Minkowski theory. As a consequence of Theorem 2, we obtain the following characterization.

Here, the map \(\operatorname{Z}\) is \(q\)-homogeneous if \(\operatorname{Z}(\lambda L)=\lambda^q\operatorname{Z}(L)\) for every \(\lambda>0\) and \(L\in\mathcal{S}_{\scriptscriptstyle 0}^n\). See Section 6.3 for the definition of dual area measures and locally determined valuations on \(\mathcal{S}_{\scriptscriptstyle 0}^n\).

We will establish our results in the setting of valuations on function spaces and identify \(\mathcal{S}_{\scriptscriptstyle 0}^n\) with \(C({S^{n-1}})_{\scriptscriptstyle +}\), the space of non-negative, continuous functions on \({S^{n-1}}\). We remark that the first classification result for valuations on function spaces was established for Sobolev functions [12], followed by results on Lebesgue spaces, spaces of continuous functions, functions of bounded variation, and convex functions (see [5]–[7], [13]–[34], LiMa? for some recent results and the survey [35]). Measure-valued valuations were considered on convex functions on \(\mathbb{R}^n\) in [24], [26]. Our results establish a complete classification of measure-valued valuations that are weak\(^*\) continuous on non-negative, continuous functions on \({S^{n-1}}\).

The rest of the paper is structured as follows: In Section 2, we present the notation and basic definitions used throughout the paper. In Section 3, we consider valuations on star bodies taking values in dual Banach spaces. Within this general framework and building on the scalar-valued case proven in [6], we derive some basic results and introduce the notion of a control measure for such valuations. In Section 4, given a weak\(^*\) continuous valuation taking values in the space of measures \(M({S^{n-1}})_{\scriptscriptstyle +}\), we construct a control measure \(\mu\). Proceeding as in [6], we establish an integral representation for such a valuation with respect to the control measure and study some of its properties. Section 5 is devoted to the proof of Theorem 1. In Section 6, we focus on valuations satisfying additional properties such as rotation equivariance, homogeneity, and local determination. There, we prove Theorems 2 and 3. In addition, we describe a valuation that is rotation equivariant and weak\(^*\) continuous but not norm continuous. Finally, in the Appendix 7, we include, for the reader’s convenience, the details of an argument which is implicit in [6].

2 Notation and Preliminaries↩︎

Let \({S^{n-1}}\) be the unit sphere of the Euclidean space \(\mathbb{R}^n\), and let \(C({S^{n-1}})\) denote the Banach space of continuous functions on \({S^{n-1}}\), endowed with the supremum norm \[\|f\|_{\scriptscriptstyle \infty}=\sup\{|f(s)|: {s\in {S^{n-1}}}\}.\] We will denote the cone of non-negative functions in \(C({S^{n-1}})\) by \(C({S^{n-1}})_{\scriptscriptstyle +}\). Let \(\Sigma_n\) be the Borel \(\sigma\)-algebra in \({S^{n-1}}\) and \(B({S^{n-1}})\) the Banach space of bounded Borel measurable functions \(g\colon{S^{n-1}}\to \mathbb{R}\), endowed with the supremum norm. We write \(B({S^{n-1}})_{\scriptscriptstyle +}\) for the cone of non-negative functions in \(B({S^{n-1}})\). Note that the formal inclusion of \(C({S^{n-1}})\) into \(B({S^{n-1}})\) is a linear isometric embedding.

2.1 Star bodies↩︎

Given \(x \in \mathbb{R}^n\), let \([0,x]=\{\lambda x:0\leq \lambda\leq 1\}\) denote the line segment joining the origin with \(x\). A set \(L\subset \mathbb{R}^n\) is a star set if \([0,x]\subset L\) for every \(x\in L\). For a star set \(L\subset \mathbb{R}^n\), the radial function \(\rho_L\colon{S^{n-1}}\to\mathbb{R}_{\scriptscriptstyle +}\) is given by \[\rho_L(s)= \sup \{\lambda\geq 0 : \lambda s\in L\}.\] A star set \(L\) is called a star body whenever \(\rho_L \in C({S^{n-1}})_{\scriptscriptstyle +}\). Conversely, for \(f\in C({S^{n-1}})_{\scriptscriptstyle +}\), there exists a star body \(L_f\) such that \(f=\rho_{L_f}\). Let \(\mathcal{S}_{\scriptscriptstyle 0}^n\) denote the set of star bodies in \(\mathbb{R}^n\). Note that star bodies are always bounded.

For \(K,L\in \mathcal{S}_{\scriptscriptstyle 0}^n\), the radial sum \(K \mathbin{\tilde{+}}L\) is defined as the star body with radial function satisfying \[\rho_{K\mathbin{\tilde{+}}L}=\rho_K+\rho_L.\] The radial metric is defined by \[\delta(K,L)=\inf\{\varepsilon\geq 0 : K\subset L\mathbin{\tilde{+}}\varepsilon B^n, L\subset K\mathbin{\tilde{+}}\varepsilon B^n\},\] where \(B^n\) denotes the Euclidean unit ball of \(\mathbb{R}^n\). It is easy to check that \[\delta(K,L)=\sup\{|\rho_K(s)-\rho_L(s)|: s\in {S^{n-1}}\}=\|\rho_K-\rho_L\|_{\scriptscriptstyle \infty}.\] We equip \(\mathcal{S}_{\scriptscriptstyle 0}^n\) with the topology induced by the radial metric.

2.2 Valuations on star bodies↩︎

Given an additive abelian semigroup \(X\), a function \(\operatorname{Z}\colon\mathcal{S}_{\scriptscriptstyle 0}^n\to X\) is a valuation if \[\label{eq:valuation} \operatorname{Z}(K\cup L)+\operatorname{Z}(K\cap L)=\operatorname{Z}(K)+ \operatorname{Z}(L)\tag{2}\] for all \(K,L\in\mathcal{S}_{\scriptscriptstyle 0}^n\).

Note that if \(K, L \in\mathcal{S}_{\scriptscriptstyle 0}^n\), then both \(K\cup L\) and \(K\cap L\) are in \(\mathcal{S}_{\scriptscriptstyle 0}^n\). It is easy to see that \[\rho_{K\cup L}=\rho_K\vee \rho_L,\rho_{K\cap L}=\rho_K\wedge \rho_L,\] where for functions \(f, g\colon{S^{n-1}}\to \mathbb{R}\) and \(s\in {S^{n-1}}\), we set \[(f\vee g)(s)=\max \{f(s), g(s)\},\] \[(f\wedge g)(s)=\min \{f(s), g(s)\}.\] Therefore, we can identify the valuation \(\operatorname{Z}\colon\mathcal{S}_{\scriptscriptstyle 0}^n\rightarrow X\) with the valuation (in the function space sense) \(\operatorname{V}\colon C({S^{n-1}})_{\scriptscriptstyle +}\rightarrow X\), with the natural identification given by \[\label{eq95star} \operatorname{V}(f)=\operatorname{Z}(L_f),\tag{3}\] where \(L_f\) is the star body with radial function \(\rho_{L_f}=f\). In the function space setting, the valuation property 2 becomes \[\label{eq:valuationfunctions} \operatorname{V}(f\vee g)+\operatorname{V}(f\wedge g)=\operatorname{V}(f)+ \operatorname{V}(g)\tag{4}\] for every \(f,g \in C({S^{n-1}})_{\scriptscriptstyle +}\).

2.3 Measure-valued valuations↩︎

General valuations are often hard to describe, so continuity assumptions are used to single out the most relevant classes of valuations. In this paper, we focus on weak\(^*\) continuous valuations on star bodies taking values in the space of measures on the sphere.

Let \(M({S^{n-1}})\) be the space of signed Radon measures on \({S^{n-1}}\), equipped with the variation norm \(\|\mu\|=|\mu|({S^{n-1}})\). The classical Riesz representation theorem (cf.[36]) states that \(M({S^{n-1}})\) can be identified with the Banach space dual of \(C({S^{n-1}})\). The duality pairing is given by \[\label{eq:pairing} \mu(f)=\langle \mu, f\rangle=\int_{{S^{n-1}}} f(s) \,\mathrm{d}\mu(s)\tag{5}\] for \(f\in C({S^{n-1}})\) and \(\mu\in M({S^{n-1}})\).

We recall that, given a Banach space \(X\), its dual \(X^*\) consists of all bounded linear functionals \(x^*\colon X\to \mathbb{R}\), and the weak\(^*\) topology in \(X^*\) is defined as the coarsest topology making continuous the evaluation on points of \(X\), in other words, the topology induced by neighborhoods of the form \[\{x^*\in X^*:\max_{1\leq i\leq m} |x^*(x_i)-x_0^*(x_i)|<\varepsilon\}\] for every \((x_i)_{i=1}^{m}\subset X\), \(x_0^*\in X^*\) and \(\varepsilon>0\).

We say that a valuation \(\operatorname{V}\colon C({S^{n-1}})\rightarrow M({S^{n-1}})\) is weak\(^{\,*}\) continuous if it is continuous when we consider the norm topology in \(C({S^{n-1}})\) and the weak\(^*\) topology in \(M({S^{n-1}})\), or, equivalently, if the scalar-valued valuation \(\operatorname{V}_g\colon C({S^{n-1}})\rightarrow \mathbb{R},\) defined by \[\operatorname{V}_g(f)=\langle \operatorname{V}(f),g\rangle=\int_{{S^{n-1}}} g(s)\,\mathrm{d}\operatorname{V}(f)(s),\] is continuous for every \(g\in C({S^{n-1}})\).

We denote the orthogonal group in dimension \(n\) by \(\operatorname{O}(n)\). Given a function \(f\colon {S^{n-1}}\to \mathbb{R}\) and \(\phi \in \operatorname{O}(n)\), we will often need the rotation of \(f\) by \(\phi\), which we will denote by \(\phi f= f\circ \phi^{-1}\).

Similarly, given \(A\in \Sigma_n\), the rotation of \(A\) by \(\phi\) corresponds to the Borel set \(\phi A=\{\phi(t)\colon t\in A\}\), which in terms of the corresponding characteristic functions \(\chi_A\) can be written as \(\phi\chi_{A} = \chi_{\phi A}\).

We say that a valuation \(\operatorname{V}\colon C({S^{n-1}})\to M({S^{n-1}})\) is rotation equivariant if \[\operatorname{V}(\phi f)(\phi B)=\operatorname{V}(f)(B)\] for every \(f\in C({S^{n-1}})\), \(B\in \Sigma_n\), and \(\phi\in \operatorname{O}(n)\); equivalently, by the Riesz representation theorem, \[\langle \operatorname{V}(\phi f), \phi g\rangle =\langle \operatorname{V}(f),g\rangle\] for every \(f,g \in C({S^{n-1}})\) and \(\phi\in \operatorname{O}(n)\).

It is convenient to work with valuations defined on the cone of non-negative, continuous functions, \(C({S^{n-1}})_{\scriptscriptstyle +}\). We remark that there is more than one way to extend a valuation \(\operatorname{V}\colon C({S^{n-1}})_{\scriptscriptstyle +}\to M({S^{n-1}})\) to a map defined on the whole space \(C({S^{n-1}})\): define \(\operatorname{V}^\pm\colon C({S^{n-1}})\to M({S^{n-1}})\) by \[\operatorname{V}^\pm(f)=\operatorname{V}(f\vee 0)\pm \operatorname{V}((-f)\vee 0).\] It is straightforward to check that \(\operatorname{V}^\pm\) also satisfies the valuation property 4 . Conversely, given a valuation \(\operatorname{V}\colon C({S^{n-1}})\to M({S^{n-1}})\), we define the maps \(\operatorname{V}_\pm\colon C({S^{n-1}})_{\scriptscriptstyle +}\to M({S^{n-1}})\) by \[\operatorname{V}_\pm(f)=\operatorname{V}(\pm f).\] It is clear that \(\operatorname{V}_\pm\) are valuations. Assuming that \(\operatorname{V}(0)=0\), we can always write \[\label{e:decomp} \operatorname{V}(f)=\operatorname{V}_+(f_{\scriptscriptstyle +})+\operatorname{V}_-(f_{\scriptscriptstyle-}),\tag{6}\] where \(f_{\scriptscriptstyle +}=f\vee 0\) and \(f_{\scriptscriptstyle-}=(-f)\vee0\).

We remark that the classification results that we will derive for valuations on \(C({S^{n-1}})_{\scriptscriptstyle +}\) can be immediately extended to classification results for valuations on \(C({S^{n-1}})\) with values in the space of signed Radon measures on \({S^{n-1}}\) such that \(\operatorname{V}(0)=0\) and \(\operatorname{V}(f)\in M({S^{n-1}})_{\scriptscriptstyle +}\) for non-negative \(f\in C(\mathbb{R}^n)\) using the above decomposition. Therefore, in this work, we will focus on valuations \(\operatorname{V}:C({S^{n-1}})_{\scriptscriptstyle +}\to M({S^{n-1}})_{\scriptscriptstyle +}\) which satisfy \(\operatorname{V}(0)=0\).

2.4 Strong Carathéodory property↩︎

The following notion appeared in [37]. Let \(\mu\in M({S^{n-1}})\) be given. A map \(K\colon \mathbb{R}_{\scriptscriptstyle +}\times {S^{n-1}}\to \mathbb{R}\) is \(\mu\)-strong Carathéodory if the mapping \(s\mapsto K(\lambda, s)\) is Borel measurable on \({S^{n-1}}\) for every \(\lambda \in\mathbb{R}_{\scriptscriptstyle +}\) and there exists a Borel set \(S_0\subset{S^{n-1}}\) with \(\mu(S_0)=\mu({S^{n-1}})\) such that for every \(s\in S_0\) the mapping \(\lambda\mapsto K(\lambda,s)\) is continuous on \(\mathbb{R}_{\scriptscriptstyle +}\). If \(K\) is \(\mu\)-strong Carathéodory on \(\mathbb{R}_{\scriptscriptstyle +}\times {S^{n-1}}\), then, for every \(f\in C({S^{n-1}})\), the mapping \[s \mapsto K(f(s), s)\] is Borel measurable.

In the sequel we will need to extend this definition to cover the case of functions \(U:\mathbb{R}_{\scriptscriptstyle +}\times {S^{n-1}}\to M({S^{n-1}})_+\) where in \(M({S^{n-1}})_+\) we consider the weak\(^*\) topology. We will say that \(U\) thus defined is \(\mu\)-strong Carathéodory if, for every \(g\in C({S^{n-1}})\), the map \(U_g \colon \mathbb{R}_{\scriptscriptstyle +}\times {S^{n-1}}\to \mathbb{R}\) given by \(U_g(\lambda, s) = U(\lambda, s)(g)\) is \(\mu\)-strong Carathéodory.

3 Vector-Valued Valuations on Star Bodies↩︎

This paper aims to classify measure-valued valuations defined on star bodies. Still, some preliminary results are better understood when stated and proved in slightly greater generality.

In this section, we will focus on valuations defined on \(C({S^{n-1}})\) with values on an arbitrary dual Banach space \(X^*\) (with the dual norm \(\Vert \cdot\Vert_*\)), that is, valuations \(V:C({S^{n-1}})\to X^*\). In the next sections we will restrict to the case \(X^*=M({S^{n-1}})\).

We say that a valuation is \(\operatorname{V}\colon C({S^{n-1}})\to X^*\) is bounded on bounded sets if, for every \(\lambda>0\), there exists \(\gamma\geq 0\) such that \[\|\operatorname{V}(f)\|_*\leq \gamma\] for every \(f \in C({S^{n-1}})\) with \(\|f\|_{\scriptscriptstyle \infty}\leq \lambda\).

By [5], every continuous, scalar-valued valuation on \(C({S^{n-1}})\) is bounded on bounded sets. It follows from this fact and the Uniform Boundedness Principle, that weak\(^*\) continuous valuations are bounded on bounded sets:

The main tools to describe and classify continuous, scalar-valued valuations on star bodies in [5]–[7] came from Measure Theory. Specifically, the Riesz representation theorem for linear functionals on \(C({S^{n-1}})\) was used several times.

A key idea in those papers was to use the fact that the values of a valuation on continuous functions which approximate pointwise a characteristic function of the form \(\lambda \chi_A\), with \(\lambda\in\mathbb{R}_{\scriptscriptstyle +}\) and \(A\in\Sigma_n\), can be used to define a family of measures \(\omega_\lambda\) on \(\Sigma_n\) which depend continuously on \(\lambda\). This is essentially the same as extending a valuation \(\operatorname{W}\colon C({S^{n-1}})\to \mathbb{R}\) to a valuation \(\overline{\operatorname{W}}\colon B({S^{n-1}})\to \mathbb{R}\) in a canonical way.

In this paper, we will use the vector-valued version of this approach. Let us start by considering scalar-valued valuations obtained by composing with elements in \(X\).

Given a weak\(^*\) continuous valuation \(\operatorname{V}\colon C({S^{n-1}})\to X^*\) and \(x\in X\), let \(\overline{\operatorname{V}}_x\colon B({S^{n-1}})\to \mathbb{R}\) be the canonical extension of \(\operatorname{V}_x=\langle \operatorname{V},x\rangle\) to the space of bounded Borel functions as constructed in [5]. For completeness, we outline the extension construction here without providing proofs of the statements. Details can be found in [5].

First, by [5], every valuation \(\operatorname{W}\colon C({S^{n-1}})\to \mathbb{R}\) can be written as a difference of non-negative valuations, that is, there exist non-negative valuations \(\operatorname{W}^+\) and \(\operatorname{W}^-\) such that \(\operatorname{W}=\operatorname{W}^+- \operatorname{W}^-\) (however, this decomposition need not be unique). The following construction defines a Radon measure for every \(\lambda > 0\) and each sign \(\pm\): For \(A\in\Sigma_n\), we set \[\label{eq:def32nu32lambda43} \omega_{\lambda}^{\pm}(A)= \inf_{\substack{\\ G \text{ open} \\ A\subset G}}\,\, \sup_{\substack{K \text{ closed}\\[1pt] K\subset G}} \inf\Big\{\operatorname{W}^{\pm}(f)\colon f\in C({S^{n-1}}),\,\chi_K\leq \frac{f}{\lambda} \leq \chi_G \Big\},\tag{7}\] and \[\label{eq:def32nu32lambda} \omega_{\lambda}=\omega^+_{\lambda}-\omega^-_{\lambda}.\tag{8}\]

For a simple function \(g=\sum_{i=1}^m \lambda_i\chi_{A_i}\) with pairwise disjoint \(A_i\in \Sigma_n\) and \(\lambda_i >0\), we define \[\overline{\operatorname{W}}(g)=\sum_{i=1}^m\omega_{\lambda_i}(A_i).\] This definition admits a unique continuous extension to \(B({S^{n-1}})\), and it can be checked that \(\overline{\operatorname{W}}(f)=\operatorname{W}(f)\) for every \(f\in C({S^{n-1}})\) (see [5]). Note that the extended valuation is continuous (with respect to the norm \(\|\cdot\|_{\scriptscriptstyle \infty}\)). In particular, the following identity will be useful (see [5]): for every closed set \(C\subset {S^{n-1}}\) and \(\lambda \ge 0\), we have \[\label{limitrims} \omega_{\lambda}(C)=\lim_{i\to \infty} \operatorname{W}(f_i)\tag{9}\] whenever \(f_i\in C({S^{n-1}})_{\scriptscriptstyle +}\) are such that \(\|f_i\|_{\scriptscriptstyle \infty}\leq \lambda\), \(f_i(s)=\lambda\) for \(s\in C\), and \(f_i(s)=0\) when \(dist(s,C)\geq1/i\) (where \(dist(s,C)\) is the infimum of the spherical distances of \(s\) and \(t\in C\)).

Next, we show that the extension procedure is linear.

In the following, we introduce scalar measures associated to a weak\(^*\) continuous valuation \(\operatorname{V}\colon C({S^{n-1}})\to X^*\) with \(\operatorname{V}(0)=0\). For \(\lambda\in \mathbb{R}_{\scriptscriptstyle +}\) and \(x\in X\), let \[\label{eq:defnulambda} \nu_{\lambda,x}= \omega_\lambda\tag{10}\] be the scalar-valued measure associated to \(\operatorname{W}=\langle\operatorname{V},x\rangle\) in 8 .

Now, given \(A\in\Sigma_n\), we consider the linear and continuous mapping \(x\mapsto \nu_{\lambda,x}(A)\). The linearity of this mapping follows from Lemma 1 and the continuity from the definition of \(\nu_{\lambda,x}(A)\) and Proposition 1. Note also, that for a fixed \(x\in X\) the mapping \(A\mapsto \nu_{\lambda,x}(A)\) is a measure on \(\Sigma_n\). We will use the following notion of control measure.

4 Construction of a control measure and further results from the scalar case↩︎

Throughout the remainder of the paper we restrict to the case where \(X=C({S^{n-1}})\) and denote by \(\|\cdot \|\) the dual norm on \(M({S^{n-1}})\), dropping the subscript \(*\). A key difference with respect to the scalar case is that while every scalar valuation can be written as the difference of two non-negative valuations, this property fails for vector-valued valuations. To see this, it suffices to consider a valuation defined by a continuous linear operator \(T:C(S^{n-1})\to M(S^{n-1})\) that is not absolutely summing (hence, it cannot be written as a difference of positive linear operators, see [39]). Thus, from now on we focus on valuations \(\operatorname{V}:C({S^{n-1}})_{\scriptscriptstyle +}\to M({S^{n-1}})_{\scriptscriptstyle +}\) that are weak\(^*\) continuous and satisfy \(\operatorname{V}(0)=0\).

In previous works [5], [6], a control measure was constructed for scalar valuations. Let us first recall this construction. Given a non-negative valuation \(\operatorname{W}:C({S^{n-1}})\to\mathbb{R}_{\scriptscriptstyle +}\) and \(\lambda_0\geq0\), a Radon measure \(\mu_{\lambda_0}:\Sigma_n\to\mathbb{R}\) can be defined such that on open sets \(G\subset {S^{n-1}}\) \[\label{eq:control32measure} \mu_{\lambda_0}(G)=\sup\{\operatorname{W}(f)\colon f\prec G, \, \|f\|_{\scriptscriptstyle \infty}\leq \lambda_0\},\tag{11}\] where \(f\prec G\) means that the support of \(f\) is contained in \(G\).

The key property of \(\mu_{\lambda_0}\), and the one needed for the arguments in the previous works [5], [6], is its “control” property: for every \(A\in\Sigma_n\) and \(f\in C({S^{n-1}})\), \[\label{eq:control} f\prec A \,\text{ and }\,\|f\|_{\scriptscriptstyle \infty}\leq\lambda_0\,\text{ imply }\,|\operatorname{W}(f)|\leq \mu_{\lambda_0}(A).\tag{12}\] Suppose that \(f\in C({S^{n-1}})_{\scriptscriptstyle +}\). Let us consider the constant one function \(\operatorname{\mathbb{1}}\). It follows from the positivity of \(\operatorname{V}\) that \[0\leq \langle \operatorname{V}(f),g\rangle\leq \langle \operatorname{V}(f),\|g\|_{\scriptscriptstyle\infty}\operatorname{\mathbb{1}}\rangle\] for \(g\ge0\). Hence, \[\label{Eq:inequality} |\operatorname{V}_g(f)|=|\langle \operatorname{V}(f),g\rangle|=|\langle \operatorname{V}(f),g_+-g_-\rangle|\leq \langle \operatorname{V}(f),|g|\rangle\leq \operatorname{V}_{\operatorname{\mathbb{1}}}(f)\,\|g\|_{\scriptscriptstyle \infty}\tag{13}\] for \(g\in C({S^{n-1}})\).

Let \(\mu_{\lambda_0,\operatorname{\mathbb{1}}}\) be the measure defined in 11 associated to the scalar-valued valuation \(\operatorname{V}_{\operatorname{\mathbb{1}}}=\langle \operatorname{V}, \operatorname{\mathbb{1}}\rangle\). It follows from 13 that for every \(g\in C({S^{n-1}})\), \(f\in C({S^{n-1}})_{\scriptscriptstyle +}\) and \(A\in\Sigma_n\) with \(\|f\|\leq \lambda_0\) and \(f\prec A\), we obtain \[|V_g(f)|\leq V_{\operatorname{\mathbb{1}}} (f)\|g\|_{\scriptscriptstyle \infty} \leq \mu_{\lambda_0,\operatorname{\mathbb{1}}} (A)\|g\|_{\scriptscriptstyle \infty}.\] In order to handle arbitrarily large functions \(f\in C({S^{n-1}})\), we will use the following measure: \[\label{eq95deK95mu} \mu=\sum_{i=1}^\infty\frac{\mu_{i,\operatorname{\mathbb{1}}}}{2^i\|\mu_{i,\operatorname{\mathbb{1}}}\|},\tag{14}\] where \(\|\mu_{i,\operatorname{\mathbb{1}}}\|\) denotes the variation norm. Note that the above series is absolutely convergent and, therefore, defines a proper, non-negative, regular measure. Although the measure \(\mu\) may be defined on a larger \(\sigma\)-algebra, we restrict to Borel sets, since this is sufficient for our purposes.

Now let us see that the measure \(\mu\) is a control measure in the sense of ?? . It follows easily that if \(f\in C({S^{n-1}})_{\scriptscriptstyle +}\), \(g\in C({S^{n-1}})\) and \(A\in\Sigma_n\) with \(f\prec A\) one has \[\label{Eq:relating} |V_g(f)|\leq \|g\|_{\scriptscriptstyle \infty} \mu_{m,\operatorname{\mathbb{1}}}(A) \leq \|g\|_{\scriptscriptstyle \infty} 2^m \|\mu_{m,\operatorname{\mathbb{1}}}\|\mu(A)\tag{15}\] for every \(m\in\mathbb{N}\) such that \(\|f\|_{\scriptscriptstyle \infty}\leq m\). Define now the function \(\zeta\) as follows \[\label{eq95cm} \zeta(t)=2^{\lceil t\rceil}\|\mu_{\operatorname{\mathbb{1}}, \lceil t\rceil}\|.\tag{16}\] where \(\lceil t\rceil=\min\{m\in\mathbb{Z}:t\leq m\}\) denotes the ceiling function.

Finally, to check that \(\mu\) is a control measure we have to verify inequality ?? . Let \(C\subset {S^{n-1}}\) be a closed set, let \(\lambda> 0\) and let \(g\in C({S^{n-1}})\). Consider a sequence \(\{f_m\}_{m\in\mathbb{N}}\subset C({S^{n-1}})_{\scriptscriptstyle +}\) with \(\|f_m\|_{\scriptscriptstyle \infty}\leq \lambda\), \(f_m(s)=\lambda\) for \(s\in C\) and \(f_m(s)=0\) whenever \(dist(s,C)\geq1/m\). By 9 , \[|\nu_{\lambda,g}(C)|=|\overline{\operatorname{V}}_g(\lambda \chi_C)|=\lim_{m\to\infty}|\operatorname{V}_g(f_m)|\leq \|g\|_{\scriptscriptstyle \infty} \zeta(\lambda)\mu(C).\] The regularity of the measures implies now that the same inequality holds for Borel sets. That is, for every \(A\in\Sigma_n\), \[|\nu_{\lambda,g}(A)|\leq \|g\|_{\scriptscriptstyle \infty} \zeta(\lambda)\mu(A).\] Hence, \(\mu\) is a control measure with \(\zeta\) as the control function.

We state for further reference the following trivial fact about the control function:

Our next objective is to adapt the integral representation result for scalar-valued valuations from [6]. In that paper, the representation was obtained for a specific control measure which does not necessarily coincide with our \(\mu\). Thus, our goal is to obtain an integral representation of \(\operatorname{V}_g\) with respect to the measure \(\mu\) defined in 14 . This representation with \(\mu\) follows directly from the proof of [6] where it becomes apparent that the only requirement on the measure \(\mu\) is the control property. In particular, one can prove the following result.

We will repeatedly rely on the fact that \(K_g(\lambda, \cdot)\) belongs not only to \(L_1({S^{n-1}},\mu)\), but also to \(L_{\scriptscriptstyle \infty}({S^{n-1}},\mu)\). This will be established in the next lemma.

5 Characterization of weak\(^*\) continuous valuations↩︎

Let \(\mathcal{P}_{\mathbb{Q}}\) be the set (of restrictions to \({S^{n-1}}\)) of polynomials with rational coefficients defined on \(\mathbb{R}^n\). It follows from the Stone–Weierstrass Theorem that \(\mathcal{P}_{\mathbb{Q}}\) is a (countable) dense subset of \(C({S^{n-1}})\). For our purposes, we could actually take any countable dense set in \(C({S^{n-1}})\), closed under sums and multiplication by rational numbers, instead of \(\mathcal{P}_{\mathbb{Q}}\). Let \(\left( p_j\right)_{j\in \mathbb{N}}\) be an enumeration of \(\mathcal{P}_{\mathbb{Q}}\) and let \((\lambda_m)_{m\in\mathbb{N}}\) be an enumeration of \(\mathbb{Q}_{\scriptscriptstyle +}\). We consider now the functions \(K_{p_j}\) as in Lemma 2.

For every pair \((p_i,p_j)\subset \mathcal{P}_{\mathbb{Q}}\) and every \(m\in \mathbb{N}\), let us consider \[S^{m}_{i,j}= \{s\in{S^{n-1}}:K_{p_i}(\lambda_m,s)+K_{p_j}(\lambda_m,s)\neq K_{p_i+p_j}(\lambda_m,s)\}.\] By Lemma 4, we obtain that \(S^{m}_{i,j}\) is Borel, \(\mu(S^m_{i,j})=0\) and, therefore, the set \(S_1=\bigcup_{i,j,m\in\mathbb{N}}S^m_{i,j}\) is also Borel with \(\mu(S_1)=0\).

Observe that for each \(s\notin S_1\) and all \(i,j,m\in\mathbb{N}\), the relation \(K_{p_i}(\lambda_m,s)+K_{p_j}(\lambda_m,s)=K_{p_i+p_j}(\lambda_m,s)\) holds.

Now, for \(\lambda\not\in\mathbb{Q}_{\scriptscriptstyle +}\), consider \(\{\lambda_m\}_{m\in\mathbb{N}}\subset\mathbb{Q}_{\scriptscriptstyle +}\) converging to \(\lambda\). Since \(K_{p}(\cdot,s)\) is continuous for every \(s\not\in S_1\) and every \(p\in\mathcal{P}_{\mathbb{Q}}\), for every \(s\not\in S_1\), \[\begin{align} K_{p_i}(\lambda,s)+K_{p_j}(\lambda,s)&=\lim_{m\to\infty}K_{p_i}(\lambda_m,s)+K_{p_j}(\lambda_m,s)\\ &=\lim_{m\to\infty}K_{p_i+p_j}(\lambda_m,s)=K_{p_i+p_j}(\lambda,s). \end{align}\] Hence, \(p_i\mapsto K_{p_i}(\lambda,s)\) is additive for every \(\lambda\geq 0\) and every \(s\not\in S_1\). It is well known that every additive function is \(\mathbb{Q}\)-linear. Therefore, \[K_{\alpha p}(\lambda,s)=\alpha K_p(\lambda,s)\] for every \(\alpha\in\mathbb{Q}\), every \(p\in\mathcal{P}_{\mathbb{Q}}\), every \(\lambda\in\mathbb{R}_{\scriptscriptstyle +}\), and every \(s\not\in S_1\).

Next, we will make a similar procedure for the boundedness of the mapping. For every \(p_i\in\mathcal{P}_{\mathbb{Q}}\) and every \(\lambda_m\in\mathbb{Q}_{\scriptscriptstyle +}\), let \[S^{m}_i= \{s\in{S^{n-1}}: K_{p_i}(\lambda_m,s) >\zeta(\lambda_m)\|p_i\|_{\scriptscriptstyle \infty} \}.\] By Lemma 3, \(S^{m}_i\) is Borel and \(\mu(S^m_i)=0\). Let \(S_2=\bigcup_{i,m\in\mathbb{N}}S^m_i\) which is also Borel and satisfies \(\mu(S_2)=0\). Again, for \(\lambda\in\mathbb{R}_{\scriptscriptstyle +}\), take a sequence \(\{\lambda_m\}_{m\in\mathbb{N}}\subset\mathbb{Q}_{\scriptscriptstyle +}\) with \(\lambda_m\to\lambda\). Since \(K_{p_i}(\cdot,s)\) is continuous, for every \(s\not\in S_2\), \[K_{p_i}(\lambda,s)=\lim_{m\to\infty}K_{p_i}(\lambda_m,s)\leq \limsup_{m\to\infty}\zeta(\lambda_m) \|p_i\|_{\scriptscriptstyle \infty}.\] By Fact 2, we have \(\limsup_{m\to\infty}\zeta(\lambda_m)\leq \zeta(\lambda)\). Thus, for every \(s\not\in S_2\) and every \(p_i\in\mathcal{P}_{\mathbb{Q}}\), we have \[K_{p_i}(\lambda,s)\leq \zeta (\lambda)\|p_i\|_{\scriptscriptstyle \infty}.\] Finally we define \[\label{eq95set} S={S^{n-1}}\setminus \left(S_1\cup S_2\right),\tag{19}\] which is Borel and has \(\mu\)-full measure.

For every \(\lambda\geq 0\) and every \(s\in S\), consider the mapping \[\begin{array}{ccl} \mathcal{P}_{\mathbb{Q}}&\longrightarrow &\mathbb{R} \\ p_i&\longmapsto &K_{p_i}(\lambda,s), \end{array}\] which is a linear and bounded with respect to the uniform norm, by the previous discussion. Therefore, we can consider its linear and continuous extension \[\label{def:U} U(\lambda,s):C({S^{n-1}})\to\mathbb{R}\tag{20}\] which satisfies the following fact.

For every \(s\in S\), we obtain a continuous linear functional such that \[U(\lambda,s)(p_i)=\langle U(\lambda,s),p_i\rangle=K_{p_i}(\lambda,s)\] for every \(p_i\in\mathcal{P}_{\mathbb{Q}}\) and every \(\lambda\geq0\). Hence, for every \(A\in\Sigma_n\) \[\nu_{\lambda,p_i}(A)=\int_AK_{p_i}(\lambda,s)\,\mathrm{d}\mu(s)=\int_A U(\lambda,s)(p_i)\,\mathrm{d}\mu(s).\]

Next, we show that a similar result holds not only for \(p_i\in\mathcal{P}_{\mathbb{Q}}\) but also for every \(g\in C({S^{n-1}})\). Specifically, we show that \(s\mapsto U(\lambda,s)(g)\), considered as an element in \(L_1({S^{n-1}},\mu)\), coincides with the Radon–Nikodym derivative of \(\nu_{\lambda,g}\), that is, \(U(\lambda, s)(g)=K_{g}(\lambda,s)\) as elements of \(L_1({S^{n-1}},\mu)\).

The previous result shows that for every \(g\in C(S^{n-1})\), the function of two variables \[(\lambda, s)\mapsto \langle U(\lambda, s),g\rangle\] is \(\mu\)-strong Carathéodory, and in particular, it is measurable. Moreover, \[|\langle U(\lambda, s),g\rangle|\leq \|g\|_{\scriptscriptstyle \infty}\, \zeta(\lambda)\] for every \(\lambda\ge 0\) and \(s\in S\). Therefore, the integral \[\int_{{S^{n-1}}} \langle U(f(s),s),g \rangle \,\mathrm{d}\mu(s)\] is well-defined for every \(f\in C({S^{n-1}})_{\scriptscriptstyle +}\).

We now have all the tools to prove our main result Theorem 1. First, we state it in the context of valuations on functions.

Clearly, from this result, Theorem 1 follows easily. In fact, the integral representation from Theorem 4 provides extra information about the original valuation. Recall that \(B({S^{n-1}})\), the space of bounded Borel functions, can be identified with a subspace of \(M({S^{n-1}})^*=C({S^{n-1}})^{**}\), with duality given by \(\langle g,\mu\rangle=\int_{{S^{n-1}}} g \,\mathrm{d}\mu\) for \(\mu\in M({S^{n-1}})\) and \(g\in B({S^{n-1}})\):

In particular, this means that measure-valued weak\(^*\) continuous valuations on star bodies can be extended to bounded Borel star sets preserving continuity. The following is an elementary observation that we include for the sake of clarity.

6 Rotation Equivariant, Homogeneous and locally determined valuations↩︎

In this section, we specialize our main result to obtain classification theorems for valuations with certain geometric properties.

6.1 Rotation equivariant valuations↩︎

Our first aim is to show that the integral representation of Theorem 4 has a special form in the rotation equivariant case. In particular, many of the technical steps in the previous sections can be simplified in this setting.

Recall the definition of \(\nu_{\lambda,g}\) in 10 and the definition of the control measure \(\mu\) in 14 . First, the rotation equivariance implies that for every \(\phi\in\operatorname{O}(n)\) and every \(g\in C({S^{n-1}})\) \[\label{eq:rotation} \nu_{\lambda,g}(A)=\langle \overline{V}(\lambda\chi_A),g\rangle=\langle \overline{V}(\lambda\chi_{\phi A}),\phi g\rangle=\nu_{\lambda,\phi g}(\phi A).\tag{21}\]

In particular, if \(g=\operatorname{\mathbb{1}}\), then \(\nu_{\lambda,\operatorname{\mathbb{1}}}\) is a rotation invariant measure. This will also be the case for the control measure \(\mu\).

The main advantage in this case is the fact that the function \(K_g(\lambda,s)\) from Lemma 2 can be replaced with another function \(K_g'(\lambda,s)\) with a more explicit description. This new function leads to a measure \(U'(\lambda,s)\) which satisfies the main Theorem 4 and is rotation equivariant. In order to construct this \(U'\), let \(p\in\mathcal{P}_{\mathbb{Q}}\), let \(K_p(\lambda,s)\) denote the function given by Lemma 2 and let \(S\) be the set defined in 19 .

From the Lebesgue Differentiation Theorem (see [40] for an extensive discussion), there exists a Borel set \(S_{\lambda,p}\subset {S^{n-1}}\) such that \(\mu({S^{n-1}}\setminus S_{\lambda,p})=0\) and \[K_{p}(\lambda,s)=\lim_{\rho\to 0}\frac{\nu_{\lambda,p}(B(s,\rho))}{\mu(B(s,\rho))}\] for every \(s\in S_{\lambda,p}\), where \(B(s,\rho)\) denotes the closed ball centered at \(s\) with radius \(\rho\). We will see that the limit is actually well-defined for every \(s\in{S^{n-1}}\) and every \(g\in C({S^{n-1}})\) and, therefore, we will define \(\langle U'(\lambda,s),g\rangle\) as that limit.

First, define \(S_\lambda= \bigcap_{p\in\mathcal{P}_{\mathbb{Q}}}S_{\lambda,p}\) and let \(g\in C({S^{n-1}})\). Note that \[\label{eq:U61U39ae} \lim_{\rho\to 0}\frac{\nu_{\lambda,g}(B(s,\rho))}{\mu(B(s,\rho))}=\langle U(\lambda,s),g\rangle\tag{22}\] for all \(s\in S_\lambda\). Indeed, let \(\{p_i\}_{i=1}^\infty\subset\mathcal{P}_{\mathbb{Q}}\) be a sequence that converges to \(g\), then \[\begin{align} \left|\limsup_{\rho\to 0}\frac{\nu_{\lambda,g}(B(s,\rho))}{\mu(B(s,\rho))} - \langle U(\lambda,s),g\rangle\right|\\ =\left|\limsup_{\rho\to 0}\frac{\nu_{\lambda,g}(B(s,\rho))}{\mu(B(s,\rho))} -\lim_{i\to\infty}\langle U(\lambda,s),p_i\rangle\right|\\ =\left|\limsup_{\rho\to 0}\frac{\nu_{\lambda,g}(B(s,\rho))}{\mu(B(s,\rho))} -\lim_{i\to\infty}\limsup_{\rho\to 0}\frac{\nu_{\lambda,p}(B(s,\rho))}{\mu(B(s,\rho))}\right|\\ =\left|\lim_{i\to\infty}\limsup_{\rho\to 0}\frac{\nu_{\lambda,g-p_i}(B(s,\rho))}{\mu(B(s,\rho))} \right|\leq \lim_{i\to\infty} \|g-p_i\|\zeta (\lambda). \end{align}\] The limit inferior is treated analogously. Therefore, the limit exists for any \(g\in C({S^{n-1}})\) and any \(s\in S_\lambda\). To extend beyond \(S_\lambda\), note that, since \(S_\lambda\) is a \(\mu\)-full measure set, \(S_\lambda\neq\emptyset\). If \(t\not\in S_\lambda\), then there exist \(\phi\in\operatorname{O}(n)\) and \(s\in S_\lambda\) such that \(\phi(t)=s\). By 21 and Lemma 7, we have \[\frac{\nu_{\lambda,g}(B(t,\rho))}{\mu(B(t,\rho))}=\frac{\nu_{\lambda,\phi g}(\phi( B(t,\rho)))}{\mu(B(t,\rho))}=\frac{\nu_{\lambda,\phi g}(B(s,\rho))}{\mu(B(s,\rho))}.\] Since the right hand side converges when \(\rho\) goes to 0, the left hand side also converges. Thus, for any \(s\in {S^{n-1}}\) and any \(g\in C({S^{n-1}})\), we define \[\label{eq:defin95U39} U'(\lambda,s)(g)= \lim_{\rho\to 0}\frac{\nu_{\lambda,g}(B(s,\rho))}{\mu(B(s,\rho))}.\tag{23}\] Clearly, the linearity of \(g\mapsto \nu_{\lambda,g}(A)\) for any \(A\in\Sigma_n\) together with the control property implies that \(U'\) is a linear and continuous operator defined on \(C({S^{n-1}})\), that is, a measure. Moreover, by 22 and Corollary 2, we get that \(U'\) and \(U\) represent the same valuation \(V\). The only property left to prove is the continuity on \(\lambda\).

In the following subsections, whenever we assume that the valuation is rotation equivariant we will be working with the specific map \(U'\).

We will make use of these properties to provide the following natural decomposition of \(U'(\lambda,s)\): Given \(e\in S^{n-1}\), we can decompose \(\mathbb{R}^n= \mathbb{R} e\times e^\perp\), where \(e^\perp\) denotes the hyperplane orthogonal to \(e\). This decomposition induces a bijection \(\vartheta_{e}\colon {S^{n-1}}\to (-1,1)\times S^{n-2} \cup \{(1,0,\ldots,0)\}\cup \{(-1,0,\ldots,0)\}\), defined by \[\label{e95bijection} (s_1, \ldots, s_n) \mapsto \begin{cases} \Big(s_1,\displaystyle \frac{s_2}{\sqrt{1-s_1^2}}, \ldots,\frac{s_{n}}{\sqrt{1-s_1^2}} \Big) &\text{ for } s_1\ne \pm1\\ (\pm 1,0,\ldots,0) &\text{ for } s_1=\pm 1, \end{cases}\tag{25}\] where \(s_1, \dots, s_{n}\) are the coordinates of the point \(s\in{S^{n-1}}\) with respect to an orthonormal basis \(\{e,e_2,\ldots,e_n\}\). The map \(\vartheta_{e}\) clearly defines a homeomorphism between \({S^{n-1}}\backslash\{\pm e\}\) and the cylinder \((-1,1) \times {S^{n-1}}\).

In the following, we denote the normalized Hausdorff measures with respect to the sphere as \(\overline{\mathcal{H}}^k\), that is, for any \(0\leq k\leq n\) (with \(k\) integer), we set \[\overline{\mathcal{H}}^{k}(A)=\frac{\mathcal{H}^{k}(A)}{\mathcal{H}^{k}(S^{k})}\] for any Borel set \(A\subset S^k\).

This lemma shows that, in the rotation equivariant case, the push-forward of \(U'(\lambda, e)\in M({S^{n-1}})_{\scriptscriptstyle +}\) by \(\vartheta_e\) is an element of \(M((-1,1)\times S^{n-2} \cup \{e\} \cup \{-e\})_{\scriptscriptstyle +}\) of the form \[\label{eq:sigma} \sigma_\lambda \otimes \overline{\mathcal{H}}^{n-2} + a_\lambda \,\delta_{e} + b_\lambda \delta_{-e}\tag{26}\] with \(a_\lambda, b_\lambda \in \mathbb{R}_{\scriptscriptstyle +}\). In particular, for every \(g\in C({S^{n-1}})\) \[\langle U'(\lambda,s),g\rangle =\int_{(-1,1)}\int_{S^{n-2}}g(\vartheta_s^{-1}(\alpha,t))\,\mathrm{d}\overline{\mathcal{H}}^{n-2}(t) \,\mathrm{d}\sigma_\lambda (\alpha) + a_\lambda g(s)+b_\lambda g(-s).\] Although this representation can be useful, one cannot ensure continuity in \(\lambda\) of the individual terms. For instance, \(a_\lambda\) need not be continuous in \(\lambda\). Moreover, it is not clear which conditions on \(\sigma_\lambda\), \(a_\lambda\) and \(b_\lambda\) should be imposed in order for them to define a weak\(^*\) valuation. To obtain a complete characterization, we consider the three terms together as a measure on the interval \([-1,1]\). To this end, for each \(s\in{S^{n-1}}\), we introduce the following operator from \(C({S^{n-1}})\) to \(C([-1,1])\): \[\operatorname{\overline{R}}g(s,\alpha)=\left\{\begin{array}{ll} g(s) &\text{if }\alpha=1, \\ g(-s) &\text{if }\alpha=-1, \\ \displaystyle\int_{S^{n-2}}g(\vartheta_s^{-1}(\alpha,t))\,\mathrm{d}\overline{\mathcal{H}}^{n-2}(t) &\text{if }\alpha\in(-1,1). \\ \end{array} \right.\] Clearly, the previous mapping is well-defined (the image is a continuous function), is linear, continuous and surjective. Note that it is a normalized Radon transform.

Theorem 2 follows immediately from this result.

6.2 Homogeneous valuations↩︎

We derive next the following corollary to Theorem 4. Let \(q\in\mathbb{R}\).

6.3 Locally determined valuations and dual area measures↩︎

We call a valuation \(\operatorname{V}\colon C({S^{n-1}})_{\scriptscriptstyle +}\to M({S^{n-1}})_{\scriptscriptstyle +}\) locally determined if \[\label{eq:locallydet} f_1\wedge \chi_B = f_2\wedge \chi_B \,\text{ implies }\, \operatorname{V}(f_1)(B)=\operatorname{V}(f_2)(B)\tag{27}\] for every open set \(B\subset {S^{n-1}}\) and \(f_1, f_2 \in C({S^{n-1}})_{\scriptscriptstyle +}\). In fact, an argument similar to [41] shows that a locally determined map that is weak\(^*\) continuous is actually a valuation in this context.

Note in particular that locally determined maps satisfy the following \[\label{eq95ld2} f\wedge \chi_B = 0 \,\text{ implies }\, \operatorname{V}(f)(B)=V(0)(B)=0\tag{28}\] for every open set \(B\subset {S^{n-1}}\) and \(f \in C({S^{n-1}})\). We will use 28 in the proof of the following lemma.

The following classification result is an immediate consequence of Theorem 5 and Lemma 11.

Requiring that the valuation is, in addition, \(q\)-homogeneous with \(q\in\mathbb{R}\), by Corollary 3 and the above we obtain the following classification.

Finally, we return to valuations on star bodies in \(\mathbb{R}^n\). We call a valuation \(\operatorname{Z}\colon \mathcal{S}_{\scriptscriptstyle 0}^n\to M({S^{n-1}})_{\scriptscriptstyle +}\) locally determined if the valuation \(f\mapsto \operatorname{Z}(L_{f})\), as defined in 3 , is locally determined on \(C({S^{n-1}})_{\scriptscriptstyle +}\).

For \(q\in\mathbb{R}\), Huang, Lutwak, Yang, and Zhang [11] introduced the \(q\)-th dual area measures of a convex body \(K\) with the origin in the interior as \[\label{eq:dam} \tilde{S}_q(K)(B)= \frac{1}{n} \int_B \rho_K^q(s) \,\mathrm{d}\mathcal{H}^{n-1}(s)\tag{29}\] for Borel sets \(B\subset {S^{n-1}}\). Note that the map \(K \mapsto \tilde{S}_q(K)\) is a valuation defined on convex bodies in \(\mathbb{R}^n\) with the origin in their interiors and taking values in \(M({S^{n-1}})_{\scriptscriptstyle +}\). Also, note that this valuation can be extended to all star bodies using the same definition as in 29 .

Theorem 3 is now an immediate consequence of Theorem 7. As a consequence of Theorem 6, we also get the following.

These measures play a relevant role in the dual Orlicz–Brunn–Minkowski theory (cf.[42], [43]).

6.4 A final remark on weak\(^*\) continuity versus norm continuity↩︎

Our purpose here is to exhibit a simple example of a weak\(^*\) continuous, rotation equivariant valuation that is not norm continuous, i.e. it is not continuous when \(M({S^{n-1}})\) is endowed with the norm topology. Take \(\tau_\lambda=\min\{\lambda,1\}\,\delta_{\cos\lambda}\) with \(\delta_\lambda\) the Dirac measure on \(\mathbb{R}\). Clearly, \(\tau_0=0\). Let us check that \(\lambda\mapsto\tau_\lambda\) is weak\(^*\) continuous. For every \(\xi\in C([-1,1])\), we have \[\langle \tau_\lambda,\xi\rangle=\min\{\lambda,1\} \xi(\cos\lambda)\] which is continuous in \(\lambda\), since both \(\xi\) and \(\lambda\mapsto\min\{\lambda,1\}\) are continuous. Therefore, by Theorem 5, the measure-valued function \(\tau\) defines a weak\(^*\) continuous, rotation equivariant valuation \(\operatorname{V}\). Let us show that it is not norm continuous when \(n=2\). We will identify \(\mathbb{R}^2\) with the complex plane and the unit sphere \(S^1\) with the interval \([0,2\pi]\), that is, any angle \(\beta\in[0,2\pi]\) is identified with \(e^{i\beta}\in S^1\). In this case note that for \(g\in C(S^1)\), \[\operatorname{\overline{R}}g(e^{is}, \cos\lambda) =\frac{1}{2}\left(g(e^{i(s+\lambda) }) +g(e^{i(s-\lambda)})\right).\] Now consider the function \[f(e^{is})=\left\{\begin{array}{ll} -s+2\pi & \text{if }s\in[0,\pi], \\ s & \text{if }s\in[\pi,2\pi], \end{array} \right.\] and for any \(m\in\mathbb{N}\) the functions \(f_m(e^{is})=f(e^{is})+\frac{1}{m}\) which are continuous. Clearly \(f_m\) converges uniformly to \(f\) and we have \(f,f_m\geq 1\). Thus, \[\langle U'(f(e^{is}),e^{is}),g\rangle =\frac{1}{2}\left(g(e^{0})+g(e^{i(2s)})\right)\] and \[\begin{align} \langle U'(f_m(e^{is}),e^{is}),g\rangle = \left\{\begin{array}{ll} \frac{1}{2}\big(g(e^{i\frac{1}{m}})+g(e^{i\left(2s-\frac{1}{m}\right)}) \big) & \text{if }s\in[0,\pi],\\[4pt] \frac{1}{2}\big(g(e^{i(2s+\frac{1}{m})})+g(e^{-i\left(\frac{1}{m}\right)}) \big) & \text{if }s\in[\pi,2\pi], \end{array} \right. \end{align}\] where we used the fact that \(g(e^{i(\beta+2\pi)})=g(e^{i\beta})\) for any \(\beta\in\mathbb{R}\). Then, for any \(g\in C(S^1)\), \[\begin{align} \langle \operatorname{V}(f),g\rangle &=\frac{1}{2}\int_0^{2\pi}\left(g(e^{0})+g(e^{i2s})\right)\,\mathrm{d}s =\pi g(e^0)+\frac{1}{2}\int_0^{2\pi} g(e^{is})\,\mathrm{d}s, \end{align}\] and \[\begin{gather} \langle \operatorname{V}(f_m),g\rangle =\frac{1}{2}\int_0^{\pi}\left(g(e^{i\frac{1}{m}})+g(e^{i\left(2s-\frac{1}{m}\right)})\right)\,\mathrm{d}s\\ +\frac{1}{2}\int_\pi^{2\pi}\left(g(e^{-i\frac{1}{m}})+g(e^{i\left(2s+\frac{1}{m}\right)})\right)\,\mathrm{d}s \\ =\frac{\pi}{2} \left(g(e^{i\frac{1}{m}})+g(e^{-i\frac{1}{m}})\right)+\frac{1}{2}\int_0^{2\pi} g(e^{is})\,\mathrm{d}s. \end{gather}\] Therefore, for any \(m\) one can take \(g_m(s)\) such that \(g_m(e^0)=1\) and \(g_m(e^{\frac{i}{m}})=g(e^{-\frac{i}{m}})=0\) and, we have \[\langle \operatorname{V}(f)-\operatorname{V}(f_m),g_m\rangle =\pi.\] Thus, \(\operatorname{V}\) is not norm continuous.

7 Proof of Lemma 2↩︎

For the sake of completeness, we include here the details of the proof of Lemma 2. The approach is the same as in [6], keeping track of the more general condition of control measure used in this paper. In fact, the same proof will show that one can obtain a similar representation for every measure which satisfies the control condition, given by Definition 1, and whose control function is non-decreasing.

Let \(\mu\) and \(\zeta\) be the control measure and control function of \(\operatorname{V}\) as defined in Section 4. For every \(g\in C({S^{n-1}})_{\scriptscriptstyle +}\), every \(\lambda\in\mathbb{R}_{\scriptscriptstyle +}\) and every \(A\in\Sigma_n\) we have \[|\nu_{\lambda,g}(A)|\leq \|g\|_{\scriptscriptstyle \infty} \zeta (\lambda)\mu (A).\] Also recall that \(\mu\) is non-negative, finite, Borel and regular and that \(\zeta\) is bounded on bounded sets and non-decreasing.

For \(g\in C({S^{n-1}})\), write \(\operatorname{V}_g=\langle \operatorname{V},g\rangle\), and denote by \(\overline{\operatorname{V}}_g\) its continuous extension to \(B({S^{n-1}})_{\scriptscriptstyle +}\) as in [6].

For each \(g\in C({S^{n-1}})_+\) and \(\lambda\in\mathbb{R}_+\), choose a Borel and non-negative representative of \(\Phi_g(\lambda{\operatorname{\mathbb{1}}})\), which we denote by \[\label{eq:def95K0} K_0(\lambda,s)=\Phi_g(\lambda{\operatorname{\mathbb{1}}})(s).\tag{30}\] We may also assume that \(|K_0(\lambda,s)|\leq \zeta (\lambda)\|g\|_{\scriptscriptstyle \infty}\), by the control property.

For \(\delta>0\), \(\lambda>0\), and \(A\in\Sigma_n\), let \[\omega_\lambda(\delta,A)=\sup\Big\{\int_A|K_0(\alpha,s)-K_0(\alpha',s')|\,\mathrm{d}\mu(s):\alpha,\alpha'\in[0,\lambda],|\alpha-\alpha'|\leq \delta \Big\},\] and we define \[\omega_\lambda(\delta)=\sup\big\{\sum_{i=1}^m\omega_\lambda(\delta,A_i):\bigcup_{i=1}^mA_i={S^{n-1}}, A_i\cap A_j=\emptyset \text{ for }i\neq j\big\}.\] First, note that the previous object is finite. Indeed, since \(K_0(\lambda,s)\geq 0\), \[\begin{align} \int_A|K_0(\alpha,s)-K_0(\alpha',s')|\,\mathrm{d}\mu(s)&\leq\nu_{g,\alpha}(A)+\nu_{g,\alpha'}(A)\\ &\leq \|g\|_{\scriptscriptstyle \infty} \mu(A) (\zeta(\alpha)+\zeta(\alpha')). \end{align}\] Hence, since \(\zeta\) is bounded on bounded sets, \[\omega_\lambda(\delta)\leq 2\|g\|_{\scriptscriptstyle \infty} \mu({S^{n-1}})\sup_{\alpha\leq \lambda}\zeta(\alpha)<\infty.\]

Finally, note that if \(g\in C({S^{n-1}})\), then \[\operatorname{V}_g=\operatorname{V}_{g^+}-\operatorname{V}_{g^-}.\] Therefore, the function defined as \[K_{g}(\lambda,s)=K_{g^+}(\lambda,s)-K_{g^-}(\lambda,s)\] will satisfy Lemma 2.

References↩︎