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Kubota-type formulas and supports of mixed measures

Daniel Hug, Fabian Mussnig, Jacopo Ulivelli

Abstract

Kubota's integral formula expresses the intrinsic volumes of a convex body as averages over its projections onto linear subspaces. In this work, we introduce a new class of Kubota-type formulas for mixed area measures adapted to rotations around a fixed axis, which encode a crucial disintegration property. Our construction is motivated by applications to valuations on convex functions. In the latter framework, we obtain corresponding statements for (conjugate) mixed Monge-Ampère measures. As a by-product, we characterize supports of mixed area and mixed Monge-Ampère measures, thereby confirming a special case of a conjecture by Schneider.

Kubota-type formulas and supports of mixed measures

Abstract

Kubota's integral formula expresses the intrinsic volumes of a convex body as averages over its projections onto linear subspaces. In this work, we introduce a new class of Kubota-type formulas for mixed area measures adapted to rotations around a fixed axis, which encode a crucial disintegration property. Our construction is motivated by applications to valuations on convex functions. In the latter framework, we obtain corresponding statements for (conjugate) mixed Monge-Ampère measures. As a by-product, we characterize supports of mixed area and mixed Monge-Ampère measures, thereby confirming a special case of a conjecture by Schneider.
Paper Structure (17 sections, 42 theorems, 191 equations)

This paper contains 17 sections, 42 theorems, 191 equations.

Table of Contents

  1. Introduction and Main Results
  2. Overview of the Paper
  3. Preliminaries
  4. Background on Convex Bodies
  5. Background on Convex Functions
  6. Supports of Mixed Area Measures
  7. Kubota-type Formulas
  8. A First Description of Supports
  9. Proofs of Theorem \ref{['thm:supp_ext']} and Corollary \ref{['cor:supp_s_nested']}
  10. New Properties of Mixed Monge--Ampère Measures
  11. Connections with Mixed Area Measures and Mixed Volumes on Kn
  12. Connections with Mixed Area Measures on K(n+1)
  13. Behavior under Orthogonal Projections
  14. Main Results for Mixed Monge--Ampère Measures
  15. Kubota-type Formulas
  16. ...and 2 more sections

Key Result

Theorem 1.1

If $1\leq j \leq n-1$ and $f\colon {\mathbb{S}^{n-1}}\to[0,\infty)$ is measurable, then for $K_1,\ldots,K_j\in{\mathcal{K}}^n$.

Theorems & Definitions (74)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 64 more