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The homogeneous decomposition of dually translation invariant valuations on Lipschitz functions on the sphere

Andrea Colesanti, Jonas Knoerr, Daniele Pagnini

Abstract

We show that every continuous and dually translation invariant valuation on the space of Lipschitz functions on the unit sphere of $\mathbb{R}^n$, $n\ge2$, can be decomposed uniquely into a sum of homogeneous valuations of degree $0$, $1$ and $2$. In particular, there does not exist any non-trivial, continuous and dually translation invariant valuation which is homogeneous of degree $3$ or higher. For the space of those of degree $0$, $1$ and $2$ we provide a description of a dense subspace.

The homogeneous decomposition of dually translation invariant valuations on Lipschitz functions on the sphere

Abstract

We show that every continuous and dually translation invariant valuation on the space of Lipschitz functions on the unit sphere of , , can be decomposed uniquely into a sum of homogeneous valuations of degree , and . In particular, there does not exist any non-trivial, continuous and dually translation invariant valuation which is homogeneous of degree or higher. For the space of those of degree , and we provide a description of a dense subspace.
Paper Structure (18 sections, 38 theorems, 205 equations)

This paper contains 18 sections, 38 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Notations and background
  3. Notations
  4. Some notions and results from representation theory
  5. Translation invariant valuations on convex bodies
  6. Lipschitz functions on $\mathrm{S}^{n-1}$
  7. Topology on $\mathrm{Lip}(\mathrm{S}^{n-1})$
  8. $\mathrm{GL}(n,\mathbb{R})$-operation
  9. Valuations on Lipschitz functions on $\mathrm{S}^{n-1}$
  10. Topology on $\mathrm{Val}(\mathrm{Lip}(\mathrm{S}^{n-1}))$
  11. Relation to translation invariant valuations on convex bodies
  12. Proof of Theorem \ref{['maintheorem:decomposition']}
  13. General considerations and proof for even valuations
  14. Proof for odd valuations
  15. Characterization of smooth valuations
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\mu\in\mathrm{Val}(\mathbb{R}^n)$; then there exist unique $\mu_0,\dots,\mu_n$, with $\mu_k\in\mathrm{Val}_k(\mathbb{R}^n)$ for every $k=0,\dots,n$, such that

Theorems & Definitions (68)

  • Theorem 1.1: McMullen decomposition, McMullenValuationsEulerType1977
  • Theorem 1.2: ColesantiEtAlclassinvariantvaluations2020, Theorem 1.2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Casselman-Wallach CasselmanCanonicalextensionsHarish1989
  • Proposition 2.4: Alesker Aleskermultiplicativestructurecontinuous2004, Proposition A.6.
  • Theorem 2.5: Hadwiger HadwigerVorlesungenuberInhalt1957
  • ...and 58 more