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Stability of the log-Brunn-Minkowski inequality in the case of many hyperplane symmetries

Karoly Boroczky, Apratim De

Abstract

In the case of symmetries with respect to n independent linear hyperplanes, a stability version of the logarithmic Brunn-Minkowski inequality and the logarithmic Minkowski inequality for convex bodies is established.

Stability of the log-Brunn-Minkowski inequality in the case of many hyperplane symmetries

Abstract

In the case of symmetries with respect to n independent linear hyperplanes, a stability version of the logarithmic Brunn-Minkowski inequality and the logarithmic Minkowski inequality for convex bodies is established.

Paper Structure

This paper contains 9 sections, 21 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. The case of unconditional convex bodies
  3. Coordinatewise product
  4. Linear images of unconditional convex bodies
  5. Proof of Theorem \ref{['logBMuncondstab']}
  6. Convex bodies and simplicial cones
  7. Some properties of Coxeter groups
  8. The proof Theorem \ref{['logBMstab']}
  9. Proof of Theorem \ref{['logMstab']}

Key Result

THEOREM 1.3

If convex bodies $K$ and $C$ in $\mathbb{R}^n$ are invariant under linear reflections $A_1,\ldots,A_n$ through $n$ hyperplanes $H_1,\ldots,H_n$ with $H_1\cap \ldots\cap H_n=\{o\}$, then with equality in either inequality if and only if $K=K_1+\ldots+ K_m$ and $C=C_1+\ldots + C_m$ for compact convex sets $K_1,\ldots, K_m,C_1,\ldots,C_m$ of dimension at least one and invariant under $A_1,\ldots,A_n

Theorems & Definitions (24)

  • CONJECTURE 1.1: Logarithmic Brunn-Minkowski conjecture
  • CONJECTURE 1.2: Logarithmic Minkowski conjecture
  • THEOREM 1.3: Böröczky, Kalantzopoulos
  • THEOREM 1.4
  • THEOREM 1.5
  • THEOREM 2.1: Saroglou
  • THEOREM 2.2
  • THEOREM 2.3
  • THEOREM 3.1: Prékopa-Leindler
  • THEOREM 3.2
  • ...and 14 more