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The local logarithmic Brunn-Minkowski inequality for bodies of revolution

Luca Iffland

Abstract

We prove the local logarithmic Brunn-Minkowski inequality for bodies of revolution. Furthermore, we give a generalization for one origin symmetric body of revolution and one body of revolution that does not need to be symmetric. Equality cases are discussed. The proof uses an operator theoretic approach together with the decomposition of spherical functions into isotypical components with respect to rotations around a fixed axis.

The local logarithmic Brunn-Minkowski inequality for bodies of revolution

Abstract

We prove the local logarithmic Brunn-Minkowski inequality for bodies of revolution. Furthermore, we give a generalization for one origin symmetric body of revolution and one body of revolution that does not need to be symmetric. Equality cases are discussed. The proof uses an operator theoretic approach together with the decomposition of spherical functions into isotypical components with respect to rotations around a fixed axis.
Paper Structure (7 sections, 22 theorems, 138 equations)

This paper contains 7 sections, 22 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The elliptic operator $\mathcal{D}_K$
  4. Proof of the spectral gap
  5. The operator on $E_0$
  6. The operator on $E_m$ for $m\geq 1$
  7. Auxiliary results

Key Result

Theorem A

Conjecture loc_log_bm holds true if $K$ is a body of revolution, i. e. invariant under rotations around a fixed axis.

Theorems & Definitions (49)

  • Conjecture 1.1
  • Conjecture 1.3: local $L^p$-Brunn-Minkowski inequality
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['local_log_bm_zonal']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 39 more