On the extendability by continuity of angular valuations on polytopes

TL;DR

This paper investigates conditions under which translation-invariant, weakly continuous valuations on polytopes can be extended continuously to all convex bodies, providing a specific criterion in a particular case.

Contribution

It offers a new criterion for extending certain valuations from polytopes to convex bodies, advancing understanding of valuation continuity and extendability.

Findings

A simple necessary and sufficient condition for extension in a special case.

Clarification of the relationship between weak continuity and full continuity.

Insights into McMullen's valuation classification.

Abstract

A classical theorem of P. McMullen describes all valuations on polytopes that are invariant under translations and weakly continuous, i.e., continuous with respect to parallel displacements of the facets of a polytope. While it is typically not difficult to check that a valuation is weakly continuous, it is not clear how to decide whether it admits a continuous extension to convex bodies. In a special case of McMullen's construction a simple necessary and sufficient condition on the initial data of such an extension is obtained.

Equations

$$\mu (K\cup L)=\mu (K)+\mu (L)-\mu (K\cap L)$$

$${\mu }_h(P)=\sum {\int }_{\ell \subset N_{F}P}h(\overline{F},\ell )d\ell ,$$

$${\mu }_{{\pi }^{\ast }f}(P)=\sum f(\overline{F})\gamma (F,P){\mathrm{vol}}_k(F),$$

$$\frac{1}{n-k+1}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{n+1}{k+1}\right)$$

$$\psi\colon\widetilde{\operatorname{Gr}}_{k}(\mathbb{R}^{n})\to\mbox{\Large$\wedge$}^{k}\mathbb{R}^{n}$$

$$\psi (E)=u_1\wedge \cdots \wedge u_k.$$

$$\mu (P)=\sum {\mathrm{Kl}}_{\mu }(\overline{F})\gamma (F,P){\mathrm{vol}}_k(F)$$

$$f(E)={\mathrm{Kl}}_{\mu }(E^{\perp }),$$

$$(n-1){\int }_{E_2^{n-1}}f(\frac{1}{\sqrt{n-1}}\sum _{i=1}^{n-1}{ϵ}_i{u}_i\wedge u_n)dϵ=\sum _{i=1}^{n-1}f(u_i\wedge u_n)$$

$$\mathrm{Val}=\underset{k=0}{\overset{n}{⨁}}{\mathrm{Val}}_k,$$

$$\Vert \mu \Vert =\underset{K\subset B}{\sup }|\mu (K)|,$$

$$\mu {|}_E={\mathrm{Kl}}_{\mu }(E){\mathrm{vol}}_k.$$

$$C^{\circ }=\{x\in {\mathbb{R}}^n:⟨x,y⟩\le 0\text{ for }x\in C\}.$$

$$\gamma (C)=\frac{{\mathrm{vol}}_{n-k-1}(S^{n-1}\cap C^{\circ })}{{\mathrm{vol}}_{n-k-1}(S^{n-k-1})},\text{where }k=\dim L(C).$$

$$T_{x}P=\mathrm{cl}\bigcup _{t>0}t(P-x)$$

$$N_{F}P=T_F{P}^{\circ }.$$

$$N(K)=\{(x,v)\in S{\mathbb{R}}^n:v\text{ is an outward unit normal at }x\in K\}.$$

$$K\mapsto {\int }_{N(K)}\omega$$

$$\mu (K)={\int }_{N(K)}\omega$$

$$w_1\wedge \cdots \wedge w_k$$

$$\{v\in {\mathbb{R}}^n:v\wedge w_1\wedge \cdots \wedge w_k=0\}=E.$$

$$\xi\wedge*\eta=\langle\xi,\eta\rangle\omega,\qquad\xi,\eta\in\mbox{\Large$\wedge$}^{k}\mathbb{R}^{n},$$

$$\psi (E^{\perp })=\ast \psi (E),E\in {\widetilde{\mathrm{Gr}}}_k({\mathbb{R}}^n),$$

$$⟨v_1\wedge \cdots \wedge v_k,w_1\wedge \cdots \wedge w_k⟩=\det (⟨v_i,w_j⟩).$$

$$|v_1\wedge \cdots \wedge v_k|=k!{\mathrm{vol}}_k(\mathrm{conv}\{0,v_1,\dots ,v_k\}).$$

$${\lambda }_1\ge \dots \ge {\lambda }_m\ge 0;$$

$${\lambda }_1\ge \dots \ge {\lambda }_m\text{and}{\lambda }_{m-1}\ge |{\lambda }_m|.$$

$${\lambda }_1=\cdots ={\lambda }_i=1\text{and}{\lambda }_{i+1}=\cdots ={\lambda }_m=0.$$

$${\lambda }_1=\cdots ={\lambda }_m=1;$$

$${\lambda }_1=\cdots ={\lambda }_{m-1}=1\text{and}{\lambda }_m=\pm 1.$$

Full text

Supported by DFG grant WA 3510/1-1.

1. Introduction

A function on the space of convex bodies in $\mathbb{R}^{n}$ is called a valuation if it satisfies

[TABLE]

whenever $K\cup L$ is convex. The prime examples of valuations on convex bodies are the intrinsic volumes, but also many other of the most central constructions in Convex Geometry are valuations. Remarkably, in many cases (1) together with a short list of inevitable properties determines these constructions uniquely. The seminal result in this direction is Hadwiger’s characterization of the intrinsic volumes [23]; see, e.g., [22, 31, 32, 33, 35, 41, 38, 39] for results in similar spirit. The recent work of Alesker on valuations [2, 3, 4, 6, 7, 8] completely transformed this classical line of research by introducing numerous new tools and opening up new directions that extend well beyond Convexity.

The most important examples of valuations on convex bodies are usually invariant under translations and continuous with respect to the Hausdorff metric. The restriction of such a valuation to polytopes satisfies (1) for polytopes, is invariant under translations, and is weakly continuous, i.e., continuous with respect to parallel displacements of the facets of a polytope. While it is typically not difficult to check that a valuation is weakly continuous, it is not clear how to decide whether it admits a continuous extension to convex bodies (see, e.g., [9, 24], [19, Problem 2.3.13], and also [26, 27] where this is a key issue):

Problem 1.1**.**

Specify (simple necessary and sufficient) conditions under which a translation-invariant, weakly continuous valuation on polytopes admits a continuous extensions to convex bodies.

Remark**.**

Since polytopes are a dense subset, an extension, if it exists, is unique. Moreover, by a theorem of Groemer [21], a continuous extension is automatically a valuation.

1.1. Statement of the theorem

According to McMullen [36], every translation-invariant, weakly continuous valuation on polytopes can be obtained by a fairly explicit geometric construction. A special case of this construction is the following. Let $F_{k,k+1}$ be the manifold of pairs $(E,\ell)$ where $E\subset\mathbb{R}^{n}$ is a linear subspace of dimension $k$ and $\ell\subset\mathbb{R}^{n}$ is a $1$-dimensional linear subspace orthogonal to $E$. For every continuous function $h\colon F_{k,k+1}\to\mathbb{C}$ define

[TABLE]

where $P\subset\mathbb{R}^{n}$ is a polytope, the sum extends over all $k$-dimensional faces $F$ of $P$, $\overline{F}$ is the translate of the affine hull of $F$ containing the origin, $N_{F}P$ is the normal cone of $P$ at the face $F$, $d\ell$ is the rotation-invariant probability measure on the space of rays in $\overline{F}^{\perp}$ emanating from the origin, and $h(\overline{F},\cdot)$ is considered as an even function on rays in $\overline{F}^{\perp}$. By the easy part of McMullen’s theorem [36], $\mu_{h}$ is a translation-invariant, weakly continuous valuation on polytopes.

It may sound surprising, but, as was shown by Alesker [9], there exist smooth functions $h$ such that $\mu_{h}$ does not admit a continuous extension to convex bodies. Later, several authors arrived at the same conclusion in very different ways [11, 24, 37]. In the special case $n=3$ and $k=1$ Alesker [9] described a subspace of smooth sections of a certain line bundle over $F_{1,2}$ that can be identified with those functions $h$ for which a continuous extensions is possible. A sufficient condition for all $n$ and $k$ was found by Hinderer, Hug, and Weil [24]: if $h$ lies in the image of a certain integral transform, then $\mu_{h}$ extends by continuity.

The simplest non-trivial case of the construction (2) is when the function $h$ does not depend on $\ell$. In this situation, the integrals are proportional to the external angle $\gamma(F,P)$ of the polytope $P$ at the face $F$,

[TABLE]

where $\pi\colon F_{k,k+1}\to\operatorname{Gr}_{k}(\mathbb{R}^{n})$, $(E,\ell)\mapsto E$ is the projection to the Grassmannian and $f\colon\operatorname{Gr}_{k}(\mathbb{R}^{n})\to\mathbb{C}$. If $f\equiv 1$ one obtains the intrinsic volumes. Other important examples of such valuations that admit a continuous extension are the Hermitian intrinsic volumes arising in Hermitian Integral Geometry [14] or more generally the constant coefficient valuations (see, e.g., [14, 19]). Moreover, valuations of type (3) are of interest in connection with the Angularity Conjecture [15, 44].

The problem of characterizing those functions $f$ for which (3) admits a continuous extension was explicitly stated by Fu [19, Problem 2.3.13]. Although (3) is arguably the simplest possible instance of (2), the results of [9, 24] provide almost no information on this problem. The purpose of this note is to resolve this situation by specifying a simple necessary and sufficient condition under which a continuous extension exists.

Throughout this article we assume that $n\geq 3$.

Theorem 1.2**.**

Let $1\leq k<n-1$ and $f\colon\operatorname{Gr}_{k}(\mathbb{R}^{n})\to\mathbb{C}$. Then $\mu_{\pi^{*}f}$ admits a continuous extension to convex bodies if and only if $f$ is the restriction of a $2$-homogeneous polynomial to the image of the Plücker embedding. Consequently, the space of such valuations has dimension

[TABLE]

and coincides with the space of constant coefficient valuations.

Remark**.**

As was already observed in [9] for $k=n-1$ there is no condition on $f$ except continuity. Theorem 1.2 directly implies the classification of angular curvature measures that has been recently obtain by the author [44].

Let us clarify what we mean by restriction to the image of the Plücker embedding in the statement of theorem. The Plücker embedding is a map

[TABLE]

from the Grassmannian of oriented $k$-planes in $\mathbb{R}^{n}$ into the $k$th exterior power of $\mathbb{R}^{n}$. If $u_{1},\ldots,u_{k}$ is a positively oriented orthonormal basis of $E$, then

[TABLE]

We call a function on the oriented Grassmannian even if it is invariant under change of orientation; even functions on the oriented Grassmannian correspond bijectively to functions on the Grassmannian $\operatorname{Gr}_{k}(\mathbb{R}^{n})$ of (unoriented) linear subspaces of $\mathbb{R}^{n}$. In particular one may restrict even functions on \mbox{\Large\wedge}^{k}\mathbb{R}^{n} to the image of the Plücker embedding to obtain functions on the Grassmannian $\operatorname{Gr}_{k}(\mathbb{R}^{n})$.

1.2. Outline of the proof

We conclude this introduction with a plan of the proof of Theorem 1.2. We denote by $\operatorname{Val}_{k}^{+}=\operatorname{Val}^{+}_{k}(\mathbb{R}^{n})$ the space of $k$-homogeneous, continuous valuations on convex bodies in $\mathbb{R}^{n}$ that are invariant under translations and even (i.e., invariant under reflections in the origin).

Definition 1.3**.**

A valuation $\mu\in\operatorname{Val}_{k}^{+}$ is a called angular if

[TABLE]

for every polytope $P\subset\mathbb{R}^{n}$. Here the sum is over the $k$-dimensional faces $F$ of $P$, $\operatorname{Kl}_{\mu}\colon\operatorname{Gr}_{k}(\mathbb{R}^{n})\to\mathbb{C}$ is the Klain function of $\mu$ (see Section 2 for its definition), $\overline{F}$ is the translate of the affine hull of $F$ that contains the origin, and $\gamma(F,P)$ is the external angle of $P$ at $F$. The subspace of angular valuations is denoted by $A_{k}\subset\operatorname{Val}_{k}^{+}$.

Clearly, if the valuation $\mu_{\pi^{*}f}$ extends by continuity to convex bodies, then $\mu_{\pi^{*}f}\in A_{k}$ and $\operatorname{Kl}_{\mu_{\pi^{*}f}}=f$. The strategy of the proof of Theorem 1.2 is to characterize the Klain functions of the elements of $A_{k}$. To this end we pass to the more manageable subspace of smooth valuations introduced by Alesker. This will result in no loss of generality, since, as we will prove in Section 3, smooth valuations are dense in $A_{k}$. By a deep theorem of Alesker, the Irreducibility Theorem [2], every smooth valuation can be represented by integration of a differential form over the normal cycle of a convex body (see Section 2 below for precise statements). This provides us with the computational framework to deduce by an variational argument the following necessary condition on the Klain function of $\mu\in A_{n-2}$. Here and in the following we will not notationally distinguish between functions on $\operatorname{Gr}_{k}(\mathbb{R}^{n})$ and even functions on unit length simple $k$-vectors of $\mathbb{R}^{n}$.

Proposition 1.4**.**

If $\mu\in A_{n-2}$, then the function $f\colon\operatorname{Gr}_{2}(\mathbb{R}^{n})\to\mathbb{R}$,

[TABLE]

satisfies

[TABLE]

for every orthonormal basis $u_{1},\ldots,u_{n}$ of $\mathbb{R}^{n}$, where $d\epsilon$ denotes the uniform probability measure on $E_{2}^{n-1}=\{-1,1\}^{n-1}$.

One may deduce a similar condition for $\mu\in A_{k}$. We will however proceed differently and first show that (5) holds if and only if $f$ is the restriction of a $2$-homogeneous polynomial to the image of the Plücker embedding and then finish the general case by an inductive argument.

2. Preliminaries

2.1. Convex geometry

Let us denote by $\mathcal{K}(\mathbb{R}^{n})$ the space of convex bodies in $\mathbb{R}^{n}$. Continuity of functions on this space will always be understood with respect to the Hausdorff metric. The convex hull of a subset $A\subset\mathbb{R}^{n}$ will be denoted by $\operatorname{conv}A$. For other basic notions from Convex Geometry not defined here we refer the reader to our standard reference [40].

2.1.1. Continuous valuations

Alesker’s book [10] and Chapter 6 of [40] are our references for this material. Let $\operatorname{Val}=\operatorname{Val}(\mathbb{R}^{n})$ denote the space of translation-invariant continuous valuations on $\mathbb{R}^{n}$. By McMullen’s decomposition theorem,

[TABLE]

where $\operatorname{Val}_{k}$ denotes the subspace of $k$-homogeneous valuations, i.e., those valuations $\mu$ satisfying $\mu(tK)=t^{k}\mu(K)$ for all $t>0$ and $K\in\mathcal{K}(\mathbb{R}^{n})$. Moreover, there is the splitting $\operatorname{Val}_{k}=\operatorname{Val}_{k}^{+}\oplus\operatorname{Val}_{k}^{-}$ into even and odd valuations with respect to reflections in the origin. The space $\operatorname{Val}$ is equipped with the topology of uniform convergence on compact subsets. In fact, $\operatorname{Val}$ becomes a Banach space with respect to the norm

[TABLE]

where $B$ is a convex body with nonempty interior.

The affine surface area is a notable exception of a geometrically relevant valuation that does not fit into this framework [16, 34]; of a different and more recent vintage are valuations invariant under the indefinite orthogonal groups [13, 17].

Let $\mu\in\operatorname{Val}_{k}^{+}$ and let $E\subset\mathbb{R}^{n}$ be a $k$-dimensional linear subspace. By Hadwiger’s characterization of volume (see, e.g., [29, Theorem 9.1.1]), the restriction of $\mu$ to $E$ is proportional to the $k$-dimensional volume on $E$,

[TABLE]

The map $\operatorname{Kl}\colon\operatorname{Val}_{k}^{+}\to C(\operatorname{Gr}_{k}(\mathbb{R}^{n}))$ is called the Klain embedding and $\operatorname{Kl}_{\mu}$ is called the Klain function of $\mu$. By a theorem of Klain [28], $\operatorname{Kl}_{\mu}=0$ implies $\mu=0$.

2.1.2. Convex cones

If $C\subset\mathbb{R}^{n}$ is a closed convex cone, then the largest linear subspace contained in $C$ is called the lineality space of $C$ and denoted by $L(C)$. The polar cone to $C$ is

[TABLE]

The external angle $\gamma(C)$ of $C$ is defined to be the fraction of $L(C)^{\perp}$ taken up by $C^{\circ}$. More precisely,

[TABLE]

It will be important for us that the external angle of $C$ does not depend on the ambient space: If $C\subset\mathbb{R}^{m}\subset\mathbb{R}^{n}$, then $\gamma(C)$ is the same whether computed in $\mathbb{R}^{m}$ or in $\mathbb{R}^{n}$.

The tangent cone of a polytope $P$ at $x\in P$ is the closed convex cone

[TABLE]

The lineality space of $T_{x}P$ is the linear subspace parallel to the affine hull the unique face $F$ of $P$ containing $x$ in its relative interior. Note also that $T_{x}P=T_{y}P$ if $x,y$ belong to the relative interior of the same face $F$; we denote this common cone by $T_{F}P$. The normal cone of $P$ at the face $F$ is

[TABLE]

The external angle of a polytope $P$ at a face $F$ is denoted $\gamma(F,P)=\gamma(T_{F}P)$.

If $\mu$ is a valuation on convex bodies in $\mathbb{R}^{n}$ and $E\subset\mathbb{R}^{n}$ is a linear subspace, we denote by $\mu|_{E}$ the restriction of $\mu$ to $E$.

Lemma 2.1**.**

Let $E\subset\mathbb{R}^{n}$ be a linear subspace. If $\mu\in A_{k}(\mathbb{R}^{n})$, then $\mu|_{E}\in A_{k}(E)$.

Proof.

This follows immediately from $\operatorname{Kl}_{\mu}|_{\operatorname{Gr}_{k}(E)}=\operatorname{Kl}_{\mu|_{E}}$ and the fact that the external angle $\gamma(F,P)=\gamma(T_{F}P)$ of $P\subset E\subset\mathbb{R}^{n}$ is the same whether computed in $E$ or $\mathbb{R}^{n}$. ∎

2.2. Smooth valuations and the normal cycle

The general linear group $GL(n)$ acts on $\operatorname{Val}$ by $(g\cdot\mu)(K)=\mu(g^{-1}K)$. A valuation $\mu\in\operatorname{Val}$ is called smooth, if the map $GL(n)\to\operatorname{Val}$, $g\mapsto g\cdot\mu$ is smooth.

Let us write $S\mathbb{R}^{n}=\mathbb{R}^{n}\times S^{n-1}$ for the sphere bundle of $\mathbb{R}^{n}$. As a set, the normal cycle of a convex body $K\subset\mathbb{R}^{n}$ is the set of outward unit normals to $K$,

[TABLE]

Using the metric projection $p_{K}\colon\mathbb{R}^{n}\to K$ (see [40]) it is not difficult to see that $(x,v)\to x+v$ is a bi-Lipschitz equivalence between $N(K)$ and the boundary of the convex body $K+B^{n}$, where $B^{n}$ is the Euclidean unit ball. We orient normal cycle such that this is map preserves orientations; our convention for the orientation of a domain $D\subset\mathbb{R}^{n}$ is that $v_{1},\ldots,v_{n-1}\in T_{p}\partial D$ is positively oriented if $\nu_{p},v_{1},\ldots,v_{n-1}$ is, where $\nu_{p}$ is the outward normal at the point $p\in\partial D$.

In the language of Geometric Measure Theory the preceding paragraph can be summarized by saying that the normal cycle of a convex body is an integral current. It was proved in [12] that for a smooth $(n-1)$-form $\omega$ on $S\mathbb{R}^{n-1}$ which is invariant under translations by elements of $\mathbb{R}^{n}$ the function

[TABLE]

defines a smooth valuation. The converse is also true and follows from Alesker’s Irreducibility theorem:

Theorem 2.2** ([5, Theorem 5.2.1]).**

Let $0\leq k\leq n-1$. If $\mu\in\operatorname{Val}_{k}^{\infty}$ then there exists a translation-invariant smooth differential form $\omega\in\Omega^{n-1}(S\mathbb{R}^{n})$ of bi-degree $(k,n-k-1)$ such that

[TABLE]

for every convex body $K\subset\mathbb{R}^{n}$.

2.3. Grassmannians and exterior powers

Here we recall the correspondence between simple $k$-vectors and $k$-planes in $\mathbb{R}^{n}$. For more information and proofs see, e.g., [18]. If $w_{1},\ldots,w_{k}$ is a basis of a linear subspace $E\subset\mathbb{R}^{n}$, then

[TABLE]

is a simple $k$-vector. A different choice of basis yields a proportional $k$-vector and

[TABLE]

A choice of orientation on $\mathbb{R}^{n}$ is equivalent to a choice unit $n$-vector $\omega$. With respect to this choice the Hodge star operator *\colon\mbox{\Large\wedge}^{k}\mathbb{R}^{n}\to\mbox{\Large\wedge}^{n-k}\mathbb{R}^{n} is defined by

[TABLE]

where the inner product $\langle\xi,\eta\rangle$ is defined by declaring $e_{i_{1}}\wedge\cdots\wedge e_{i_{k}}$, $1\leq i_{1}<\cdots<i_{k}\leq n$, for the the standard orthonormal basis $e_{1},\ldots,e_{n}$ of $\mathbb{R}^{n}$, to be an orthonormal basis of \mbox{\Large\wedge}^{k}\mathbb{R}^{n}. Hence, if the orientation on $E^{\perp}$ is chosen such that $E\oplus E^{\perp}=\mathbb{R}^{n}$ has orientation $\omega$, then

[TABLE]

where $\psi$ denotes the Plücker embedding (4).

Finally, let us remark that if $v_{1}\wedge\cdots\wedge v_{k}$ and $w_{1}\wedge\cdots\wedge w_{k}$ are simple $k$-vectors then

[TABLE]

In particular,

[TABLE]

2.4. Representation theory of the orthogonal group

We recall from [20] the parametrizations of the irreducible complex representations of the special orthogonal groups $SO(n)$. The description of the representations of these groups is sensitive to the parity of the dimension $n$. The irreducible representations of $SO(2m+1)$ are parametrized by integer sequences $(\lambda_{1},\ldots,\lambda_{m})$ with

[TABLE]

the irreducible representations of $SO(2m)$ are parametrized by integer sequences $(\lambda_{1},\ldots,\lambda_{m})$ with

[TABLE]

For example, the representation \mbox{\Large\wedge}^{i}V, where $V=\mathbb{C}^{n}$ is the standard representation of $SO(n)$, is irreducible for $1\leq i<m$ and corresponds to

[TABLE]

When $n$ is odd, then also \mbox{\Large\wedge}^{m}V is irreducible and has highest weight

[TABLE]

but when $n$ is even \mbox{\Large\wedge}^{m}V is not irreducible: the eigenspaces of the Hodge star operator are the irreducible subrepresentations with highest weights

[TABLE]

2.4.1. Spherical harmonics

Let $\mathcal{H}_{p}^{n}$ denote the subspace of $p$-homogeneous complex-valued polynomials on $\mathbb{R}^{n}$ that are harmonic, i.e., satisfy $\Delta f=0$. The restriction of $\mathcal{H}_{p}^{n}$ to $S^{n-1}$ yields precisely the irreducible subrepresentations of $C(S^{n-1})$ under the action of the orthogonal group. Since $\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}=4\frac{\partial^{2}}{\partial z\partial\bar{z}}$, the polynomial

[TABLE]

is in $\mathcal{H}_{p}^{n}$ and a short computation shows that it is in fact a highest weight vector of weight $(p,0,\ldots,0)$, see, e.g., [42, Exercise 7.2].

2.4.2. Restriction to the image of the Plücker embedding

Lemma 2.3**.**

The subspace of $C(\widetilde{\operatorname{Gr}}_{k}(\mathbb{R}^{n}))$ of restrictions of $2$-homogeneous polynomials on \mbox{\Large\wedge}^{k}\mathbb{R}^{n} to $\widetilde{\operatorname{Gr}}_{k}(\mathbb{R}^{n})$ decomposes under the natural action of $SO(n)$ precisely into the irreducible representations with highest weights $(2m_{1},2m_{2},\ldots,2m_{\lfloor n/2\rfloor})$ satisfying $|m_{1}|\leq 1$ and $m_{i}=0$ for $i>\min(k,n-k)$. Its dimension is

[TABLE]

Proof.

This is a well-known fact that is proved for example in [44, Lemma 4.3] and [44, Lemma 4.2] ∎

3. Reduction to smooth valuations

Lemma 3.1**.**

$A_{k}\subset\operatorname{Val}_{k}^{+}$* is closed and invariant under the action of $O(n)$.*

Proof.

If $(\mu_{i})_{i=1}^{\infty}$ is a sequence of angular valuations converging uniformly on compact subsets to some valuation $\mu\in\operatorname{Val}_{k}^{+}$, then also $\operatorname{Kl}_{\mu_{i}}\to\operatorname{Kl}_{\mu}$ uniformly. If $P$ is a polytope, then

[TABLE]

for every $i$. Passing to the limit, we see that $\mu$ is angular as well. It is clear form the definition that for every $g\in O(n)$ the valuation $g\cdot\mu$ is angular if $\mu$ is. ∎

Let us recall some terminology from representation theory (see, e.g., [30, 42]). Let $K\times V\to V$ be a representation of a group $K$ on an arbitrary vector space $V$. A vector $v\in V$ is a called $K$-finite the span of $\{k\cdot v\colon k\in K\}$ is finite-dimensional. For a compact Lie group $K$, let $\widehat{K}$ be the set of equivalence classes of irreducible finite-dimensional representations of $K$. A continuous representation of $GL(n)$ on a Frechét space $V$ is called admissible if for each $\gamma\in\widehat{O(n)}$ the sum of all irreducible $O(n)$-invariant subspaces that are in the class $\gamma$ has finite dimension. The existence of the Klain embedding $\operatorname{Kl}\colon\operatorname{Val}_{k}^{+}\to C(\operatorname{Gr}_{k}(\mathbb{R}^{n}))$ implies that the representation of $GL(n)$ on $\operatorname{Val}_{i}^{+}$ is admissible [1, Proposition 4.1].

Lemma 3.2**.**

$A_{k}\cap\operatorname{Val}_{k}^{\infty}$* is a dense subspace of $A_{k}$.*

Proof.

The action of $O(n)$ on $A_{k}$ is a continuous representation of a compact Lie group on a Banach space. It is well known that in this situation the subspace of $O(n)$-finite vectors is dense (see, e.g., [42, Theorem 3.46]). It is another well-known fact that for every admissible representation of $GL(n)$ on a Frechét space, every $O(n)$-finite vector is smooth (see, e.g., [30, Proposition 8.5]). We conclude that smooth valuations in $A_{k}$ form a dense subspace. ∎

4. Proof of Proposition 1.4

Throughout this section let $\mu\in\operatorname{Val}_{n-2}^{+}(\mathbb{R}^{n})$ be fixed. Let us also fix an orientation $or$ of $\mathbb{R}^{n}$ and let us suppose that $\mu$ is given by a translation-invariant smooth differential form $\omega\in\Omega^{n-1}(S\mathbb{R}^{n})$ of bi-degree $(n-2,1)$,

[TABLE]

Consider the simplex $S\subset\mathbb{R}^{n}$ spanned by the points $p_{0}=0$, $p_{i}=v_{i}$, $i=1,\ldots,n-1$, $p_{n}=tv_{n}$, where $t$ is a positive number, and $v_{1},\ldots,v_{n}$ is some orthonormal basis of $\mathbb{R}^{n}$. Let us make the dependence of the simplex $S$ on the choice of orthonormal basis more explicit by writing $S=S(v_{1},\ldots,v_{n})$. The first goal of this section is to prove the following lemma. Here and in the following for every function $f\colon\operatorname{Gr}_{k}(\mathbb{R}^{n})\to\mathbb{C}$ and every simple nonzero $k$-vector $w_{1}\wedge\cdots\wedge w_{k}$ we will often write $f(w_{1}\wedge\cdots\wedge w_{k})$ instead of $f(E)$, $E=\{v\in\mathbb{R}^{n}\colon v\wedge w_{1}\wedge\cdots\wedge w_{k}=0\}$.

Lemma 4.1**.**

[TABLE]

We will introduce now some notation that will be useful not only in the proof of Lemma 4.1, but also in the proof of Lemma 4.3 below. Let $F_{0},F_{1},\ldots,F_{n}$ be the facets of $S$, where the notation is chosen such that $p_{i}\notin F_{i}$. Note that every $(n-2)$-face of $S$ is of the form

[TABLE]

The dihedral angle $\theta_{ij}\in(0,\pi)$ of this face is given by

[TABLE]

where $u_{i}$ is the facet normal of $F_{i}$. Note that

[TABLE]

and

[TABLE]

Lemma 4.2**.**

Let $v_{1},\ldots,v_{n}$ be an orthonormal basis of $\mathbb{R}^{n}$. Then

[TABLE]

Proof.

Note that

[TABLE]

where $h$ is the distance from $v_{n}$ to the affine subspace spanned by $v_{1},\ldots,v_{n-1}$. Since

[TABLE]

where the first vector is in the affine hull of $v_{1},\ldots,v_{n-1}$ and the second vector is orthogonal to that affine subspace, we get

[TABLE]

Induction on $n$ finishes the proof. ∎

We divide the set of all $(n-2)$-faces of $S$ into four disjoint subsets.

1. (1)

$F_{i}\cap F_{n}$ for $i=1,\ldots,n-1$. These faces do not depend on the parameter $t$ and have volume

[TABLE]

The normal cones

[TABLE]

are constant and $\theta_{in}=\pi/2$. 2. (2)

$F_{i}\cap F_{j}$ for distinct $i,j=1,\ldots n-1$. These faces depend on $t$, but their affine hull does not. Their volume is

[TABLE]

The normal cones

[TABLE]

are constant and $\theta_{ij}=\pi/2$ 3. (3)

$F_{0}\cap F_{n}$. This face does not depend on the parameter $t$, but its normal cone does. By Lemma 4.2 we have

[TABLE]

The normal cone is

[TABLE]

and the dihedral angle is

[TABLE]

Moreover

[TABLE]

where

[TABLE] 4. (4)

$F_{0}\cap F_{i}$ for $i=1,\ldots,n-1$. These faces have nonconstant volume, affine hull, normal cone, and dihedral angle. It follows from a decomposition similar to (12) that

[TABLE]

The normal cone is

[TABLE]

and the dihedral angle is

[TABLE]

Moreover

[TABLE]

where

[TABLE]

Proof of Lemma 4.1.

Observe that

[TABLE]

where the sum is over all $(n-2)$-faces of $S$. Using the preceding division of the $(n-2)$-faces of $S$, up to an additive constant that does not depend on $t$ we have that $\mu(S(v_{1},\ldots,v_{n}))$ equals $S_{1}+S_{2}+S_{3}$, where

[TABLE]

Here we always write $\bar{x}=(0,x)\in\mathbb{R}^{n}\oplus\mathbb{R}^{n}$ and $x=(x,0)\in\mathbb{R}^{n}\oplus\mathbb{R}^{n}$ if we form wedge products. Moreover, we have used (7) to obtain

[TABLE]

and

[TABLE]

for $i=1,\ldots,n-1$.

Now we compute the first derivative with respect to $t$ and let $t\to 0^{+}$. Let us make the dependence of $S_{1}(t)$ on the choice of orthonormal basis explicit by writing $S_{1}(t)(v_{1},\ldots,v_{n})$. We have

[TABLE]

Next we use

[TABLE]

as $t\to 0^{+}$. Since the integrand does not depend on $t$, taking derivatives yields

[TABLE]

Now

[TABLE]

where $R$ contains all terms that are multiples of $v_{i}\wedge\bar{v}_{i}$ for some $1\leq i\leq n-1$. After integration over $E_{2}^{n-1}$ the terms coming from $R$ cancel and we get

[TABLE]

Finally, since

[TABLE]

we get

[TABLE]

where

[TABLE]

where $\widehat{v_{k}}$ means that this factor is omitted. If we integrate over $E_{2}^{n-1}$, then the first summand will cancel the contributions from $S_{2}(t)$ and the second summand will vanish. Indeed, the term

[TABLE]

where $1\leq j\leq n-1$, $j\neq i$, will not change sign if we replace $v_{j}$ by $-v_{j}$. Similarly after integration the third summand will vanish since

[TABLE]

where $1\leq k<n$, $k\neq i$, will not change sign if we replace $v_{k}$ by $-v_{k}$. ∎

Lemma 4.3**.**

If $\mu$ is angular, then

[TABLE]

Proof.

If $\mu$ is angular then, up to an additive constant which does not depend on $t$, we have that $\mu(S(v_{1},\ldots,v_{n-1},v_{n}))$ equals $A_{1}+A_{2}+A_{3}$, where

[TABLE]

Taking the first derivative, letting $t\to 0^{+}$, and integrating over $E_{2}^{n-1}$, the term $A_{1}(t)$ yields

[TABLE]

For $A_{2}(t)$ we use (13) and get

[TABLE]

For $A_{3}(t)$ we see as in the proof of Lemma 4.1 that after integration over $E_{2}^{n-1}$ the first derivative of

[TABLE]

vanishes. Using (14) we are left with

[TABLE]

To simplify (15) note that

[TABLE]

and hence (15) equals

[TABLE]

∎

Proof of Proposition 1.4.

Let us first assume that $\mu\in A_{n-2}$ is smooth. From Lemma 4.1 and Lemma 4.3 we conclude that

[TABLE]

for every orthonormal basis $u_{1},\ldots,u_{n}$ of $\mathbb{R}^{n}$. Applying the Hodge star operator yields

[TABLE]

Combining this with (6) we obtain (5). Since by Lemma 3.2 any continuous angular valuation can be approximated by smooth angular valuations, the claim follows. ∎

5. Necessary condition for $k=n-2$.

In the section we will deduce the necessary condition in Theorem 1.2 in the special case $k=n-2$.

Proposition 5.1**.**

Let $f\colon\operatorname{Gr}_{2}(\mathbb{R}^{n})\to\mathbb{R}$ be a continuous function. Then $f$ satisfies the relation (5) for every orthonormal basis $u_{1},\ldots,u_{n}$ of $\mathbb{R}^{n}$ if and only if $f$ is the restriction of a $2$-homogeneous polynomial on \mbox{\Large\wedge}^{2}\mathbb{R}^{n} to the image of the Plücker embedding.

Proof.

Let us first prove that the restriction of every to $2$-homogeneous polynomial satisfies (5). To see this recall that if $p\colon V\to\mathbb{C}$ is a homogeneous polynomial of degree $k$ on a finite-dimensional real vector space $V$, then there exists a symmetric multilinear function $p\colon V^{k}\to\mathbb{C}$, called the polarization of $p$ and denoted by again by $p$, which is uniquely determined by the equation

[TABLE]

For a proof see, e.g., [25, Appendix A]. Now if $f$ is the restriction of a $2$-homogeneous polynomial $p$ on \mbox{\Large\wedge}^{2}\mathbb{R}^{n} to the image of the Plücker embedding, then expanding the left-hand side of (5) by the bilinearity of the polarization of $p$ immediately yields the claim.

To prove the converse, let $W\subset C(\operatorname{Gr}_{2}(\mathbb{R}^{n}))$ be the subspace of all continuous functions $f$ on $\operatorname{Gr}_{2}(\mathbb{R}^{n})$ with the property that (5) holds for every orthonormal basis $u_{1},\ldots,u_{n}$. We have to show that $W$ is contained in the space of restrictions of $2$-homogeneous polynomials on \mbox{\Large\wedge}^{k}\mathbb{R}^{n} to $\operatorname{Gr}_{2}(\mathbb{R}^{n})$.

Clearly, the subspace $W$ is closed and invariant under the action of $SO(n)$. Hence $W$ decomposes into certain irreducible subrepresentations. The highest weights occurring in the representation $C(\operatorname{Gr}_{2}(\mathbb{R}^{n}))$ are well known. According to Strichartz [43], if $n\geq 4$, they all have multiplicity $1$ and are of the form $(2m_{1},\ldots,2m_{2},0\ldots,0)$ with integers satisfying

[TABLE]

By Lemma 2.3, the proof for $n\geq 4$ will be finished if we can show that only the highest weights with

[TABLE]

can occur in $W$. For this it will suffice to show that for all other highest weights the corresponding highest weight vector $f_{m_{1},m_{2}}$ is not an element of $W$. This step will be based on an explicit description of the highest weight vectors given by Strichartz [43] (see also [3]).

For any subspace $E\in\operatorname{Gr}_{2}(\mathbb{R}^{n})$ choose an orthonormal basis $X^{1},X^{2}$ of $E$ and consider the corresponding $n\times 2$ matrix

[TABLE]

of coordinates with respect to the standard basis $e_{1},\ldots,e_{n}$ of $\mathbb{R}^{n}$. Let $X_{j}$ denote the $j$th row of this matrix. For $l\leq\lfloor n/2\rfloor$ let $A(l)$ be the $l\times 2$ matrix whose $j$th row is $X_{2j-1}+\sqrt{-1}X_{2j}$, $j=1,\ldots,l$. Note that the $l\times l$ matrix $A(l)A(l)^{t}$ is independent of the choice of the orthonormal basis of $E$.

A highest weight vector of the irreducible subrepresentation of $C(\operatorname{Gr}_{2}(\mathbb{R}^{n}))$ with highest weight $(2m_{1},2m_{2},0,\ldots,0)$ is given by

1. (1)

if $m_{2}\geq 0$,

[TABLE] 2. (2)

if $m_{2}<0$,

[TABLE]

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

[TABLE]

where $c=\cos\phi$ and $s=\sin\phi$ for brevity.

For this choice of orthonormal basis let us evaluate (5) for a highest weight vector $f=f_{m_{1},m_{2}}$. Throughout we assume that $m_{1}\neq 0$. A straightforward computation shows that

[TABLE]

and

[TABLE]

provided $m_{2}\geq 0$; if $m_{2}<0$, then the second factor inside the integral has to be conjugated and $m_{2}$ replaced by $|m_{2}|$. We will distinguish the cases $m_{2}=0$, $m_{2}$ even (and different from [math]), $m_{2}$ odd.

Let us start with the case $m_{2}=0$. Note that by (16) and (17) both sides of (5) are trigonometric polynomials of degree $2m_{1}$. Comparing the coefficients of the leading term $c^{2m_{1}}$, we get

[TABLE]

This can only hold if $m_{1}=1$.

Next let us treat the case $m_{2}$ even and different from [math]. Again by (16) and (17) the leading term on both sides is $c^{2m_{1}}$ and comparing coefficients we get

[TABLE]

This is equivalent to

[TABLE]

Since the left-hand side is strictly less than $1$ and the right-hand side greater equal $1$, this is a contradiction. Hence no highest weight vector $f=f_{m_{1},m_{2}}$ with $m_{2}\neq 0$ even can satisfy (5).

Let us finally consider the case $m_{2}$ odd. Expanding the two factors inside the sum in (17) we get

[TABLE]

and

[TABLE]

where $\cdots$ denotes lower order terms. Observe that

[TABLE]

Integrating over $E_{2}^{n-1}$ we see that the leading term $c^{2m_{1}-2}$ on the left-side of (5) results from the product of the second terms in the above expansions and the product of the first in (18) with the third in (19). Comparing coefficients we get

[TABLE]

Consequently, $(m_{2}-1)/2$ must be odd and in particular $m_{2}\geq 3$. If $n=2k+2$ is even, then

[TABLE]

We conclude that $2^{m_{1}}$ divides $m_{1}$, a contradiction. If $n=2k+1$ is odd ($k\geq 2$), then

[TABLE]

where we put $a=m_{1}-m_{2}-1$. Since $2l\leq k^{l}$ for integers $k,l\geq 2$, we have

[TABLE]

which contradicts (20). Therefore if $m_{2}$ is odd and (5) holds for $f_{m_{1},m_{2}}$, then we must have $m_{1}-|m_{2}|=0$.

Now if $m_{1}=|m_{2}|\geq 3$ is odd, then after integration over $E_{2}^{n-1}$ we see that the leading terms in (17) and (16) are $(cs)^{m_{1}-1}$ and $c^{2m_{1}-2}$. Comparing coefficients we get

[TABLE]

For the signs to match we must have $m_{1}=4k+3$ with $k=0,1,2,\ldots$. Hence

[TABLE]

which implies that $n-1$ must contain the prime factor $2$ more then once. But then (21) can only hold if $k=0$ and $n-1=4$.

Now for $n=5$ and $m_{1}=|m_{2}|=3$ one immediately checks that (17) and (16) do coincide. Hence we are going to choose a different orthonormal basis of $\mathbb{R}^{n}$:

[TABLE]

where $a=\cos\psi,b=\sin\psi$ for brevity. A straightforward computation shows that

[TABLE]

and

[TABLE]

Comparing the coefficients the leading term we get a contradiction for $m_{1}>1$. We conclude that the validity of (5) for a highest weight vector $f_{m_{1},m_{2}}$ implies $m_{1}\leq 1$.

It remains to treat the case $n=3$. Taking orthogonal complements, we have to show that if an even continuous function $f\colon S^{2}\to\mathbb{C}$ satisfies

[TABLE]

for every pair of orthonormal vectors $u_{1},u_{2}$, then $f$ is the restriction of a $2$-homogeneous polynomial to $S^{2}$. According to (10) the function $f=f_{m_{1}}=(x_{1}-ix_{2})^{2m_{1}}$, is contained in the irreducible subrepresentation with highest weight $(2m_{1})$. Now if $m_{1}>0$, then choosing as before $u_{1}=se_{1}-ce_{3},u_{2}=e_{2}$, evaluating both sides, and comparing the coefficients of the leading term $s^{2m_{1}}$, we get

[TABLE]

Hence also in this case $m_{1}\leq 1$, as desired. ∎

Corollary 5.2**.**

If $\mu\in A_{n-2}(\mathbb{R}^{n})$, then the Klain function of $\mu$ is the restriction of a $2$-homogeneous polynomial on \mbox{\Large\wedge}^{n-2}\mathbb{R}^{n} to the image of the Plücker embedding.

Proof.

Proposition 1.4, Proposition 5.1, and (6) imply that if $\mu\in A_{n-2}$, then the Klain function of $\mu$ is the restriction of a $2$-homogeneous polynomial on \mbox{\Large\wedge}^{n-2}\mathbb{R}^{n} to the image of the Plücker embedding.

∎

6. Proof of Theorem 1.2

In this final section we provide an inductive argument that allows us to deduce Theorem 1.2 from the special case $k=n-2$ already established in Corollary 5.2. We will need the following fact.

Proposition 6.1** ([44, Proposition 4.4]).**

Let $1\leq k<n-1$ and $f$ be an even continuous function on $\widetilde{\operatorname{Gr}}_{k}(\mathbb{R}^{n})$. Suppose that for every nonzero $v\in\mathbb{R}^{n}$ there exists a $2$-homogeneous polynomial $q_{v}$ on \mbox{\Large\wedge}^{k}v^{\perp} such that $f=q_{v}$ on \widetilde{\operatorname{Gr}}_{k}(v^{\perp})\subset\mbox{\Large\wedge}^{k}v^{\perp}. Then there exists a globally defined $2$-homogeneous polynomial $q$ on \mbox{\Large\wedge}^{k}\mathbb{R}^{n} such that $f=q$ on \widetilde{\operatorname{Gr}}_{k}(\mathbb{R}^{n})\subset\mbox{\Large\wedge}^{k}\mathbb{R}^{n}.

Corollary 6.2**.**

Let $1\leq k<n-1$ and $f$ be an even continuous function on $\widetilde{\operatorname{Gr}}_{k}(\mathbb{R}^{n})$. Suppose that for every linear subspace $E\subset\mathbb{R}^{n}$ of dimension $k+2$ there exists a $2$-homogeneous polynomial $q_{E}$ on \mbox{\Large\wedge}^{k}E such that $f=q_{E}$ on \widetilde{\operatorname{Gr}}_{k}(E)\subset\mbox{\Large\wedge}^{k}E. Then there exists a globally defined $2$-homogeneous polynomial $q$ on \mbox{\Large\wedge}^{k}\mathbb{R}^{n} such that $f=q$ on \widetilde{\operatorname{Gr}}_{k}(\mathbb{R}^{n})\subset\mbox{\Large\wedge}^{k}\mathbb{R}^{n}.

Proof.

Put $m=n-k-2$. If $m=0$, then there is nothing to prove. Assume now that $m>0$ and that the theorem has been proved for $m-1$. Then for every nonzero vector $v\in\mathbb{R}^{n}$ there exists a $2$-homogeneous polynomial $q_{v}$ on \mbox{\Large\wedge}^{k}v^{\perp} such that $f=q_{v}$ on \widetilde{\operatorname{Gr}}_{k}(v^{\perp})\subset\mbox{\Large\wedge}^{k}v^{\perp}. By Proposition 6.1 this implies the existence of a globally defined $2$-homogeneous polynomial $q$ on \mbox{\Large\wedge}^{k}\mathbb{R}^{n} such that $f=q$ on \widetilde{\operatorname{Gr}}_{k}(\mathbb{R}^{n})\subset\mbox{\Large\wedge}^{k}\mathbb{R}^{n}. ∎

Proof of Theorem 1.2.

If $\mu=\mu_{\pi^{*}f}$ admits a continuous extension to convex bodies then $\mu\in A_{k}$. For every linear subspace $E\subset\mathbb{R}^{n}$ the restriction $\mu|_{E}$ of $\mu$ to $E$ is according to Lemma 2.1 angular. The Klain function of $\mu|_{E}$ is the restriction of the Klain function of $\mu$ to $\operatorname{Gr}_{k}(E)$. If the dimension of $E$ is $k+2$, then $f|_{\operatorname{Gr}_{k}(E)}=\operatorname{Kl}_{\mu|_{E}}$ is by Corollary 5.2 the restriction of a $2$-homogeneous polynomial on \mbox{\Large\wedge}^{k}E to the image of the Plücker embedding. By Corollary 6.2, $f$ is the restriction of $2$-homogeneous polynomial on \mbox{\Large\wedge}^{k}\mathbb{R}^{n} to the image of the Plücker embedding.

The converse follows from the fact proved in [44, Theorem 1.4] that to each restriction of a $2$-homogeneous polynomial to the image of the Plücker embedding corresponds a constant coefficient curvature measure. Globalizing this curvature measure we get an element of $A_{k}$. That $A_{k}$ has the stated dimension is now immediate from Lemma 2.3. ∎