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Valuations in affine convex geometry

Jakob Henkel, Thomas Wannerer

Abstract

In convex geometry, the constructions that assign to a convex body its difference body, projection body, or volume have the following properties: They are (1) invariant under volume-preserving linear changes of coordinates; (2) continuous; (3) finitely additive, and the resulting convex bodies are subsets of an irreducible representation of the special linear group. In this paper we explore the question whether there exist other constructions with these properties. We discover a surprising dichotomy: There are no new examples if one assumes translation invariance, but a plethora of examples without this assumption.

Valuations in affine convex geometry

Abstract

In convex geometry, the constructions that assign to a convex body its difference body, projection body, or volume have the following properties: They are (1) invariant under volume-preserving linear changes of coordinates; (2) continuous; (3) finitely additive, and the resulting convex bodies are subsets of an irreducible representation of the special linear group. In this paper we explore the question whether there exist other constructions with these properties. We discover a surprising dichotomy: There are no new examples if one assumes translation invariance, but a plethora of examples without this assumption.
Paper Structure (28 sections, 54 theorems, 154 equations)

This paper contains 28 sections, 54 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. General background
  3. Our results
  4. Comparison with other work
  5. Organization
  6. Acknowledgments
  7. Convex geometry
  8. Gauss curvature and affine geometry
  9. Conormal cycle
  10. Valuations
  11. Complex-valued valuations
  12. Minkowski valuations
  13. Representation theory of the general linear group
  14. Rational representations
  15. Weight spaces
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\Phi\colon \mathcal{K}(V)\to \mathcal{K}(W)$ be an invariant continuous Minkowski valuation and assume in addition that $\Phi$ is translation-invariant.

Theorems & Definitions (92)

  • Theorem 1.1: Ludwig:Minkowski
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Lemma 2.1: Hug:Contributions
  • Lemma 2.2: Hug:Contributions
  • Lemma 2.3: Hug:Contributions
  • ...and 82 more