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A strengthening of the dimensional Brunn-Minkowski conjecture implies the (B)-Conjecture

Sotiris Armeniakos, Jacopo Ulivelli

Abstract

We prove that if a sufficiently regular even log-concave measure satisfies a certain stronger form of the dimensional Brunn-Minkowski conjecture, then it also satisfies the (B)-conjecture. Furthermore, we show that hereditarily convex measures satisfy the aforementioned strengthened form, therefore providing an alternative proof of a recent result by Cordero-Erausquin and Eskenazis stating that a hereditarily convex measure satisfies both conjectures.

A strengthening of the dimensional Brunn-Minkowski conjecture implies the (B)-Conjecture

Abstract

We prove that if a sufficiently regular even log-concave measure satisfies a certain stronger form of the dimensional Brunn-Minkowski conjecture, then it also satisfies the (B)-conjecture. Furthermore, we show that hereditarily convex measures satisfy the aforementioned strengthened form, therefore providing an alternative proof of a recent result by Cordero-Erausquin and Eskenazis stating that a hereditarily convex measure satisfies both conjectures.
Paper Structure (5 sections, 4 theorems, 35 equations)

This paper contains 5 sections, 4 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and proofs of the results
  3. Proof of Theorem \ref{['thm:main_1']}.
  4. Proof of Theorem \ref{['thm:main_2']}.
  5. Acknowledgments.

Key Result

Theorem 1.1

Let $u \in C^2(\mathbb{R}^n)$ be an even convex function such that its Hessian matrix $\nabla^2 u$ is positive definite, and let $\mu$ be the associated log-concave measure with density $\, \mathrm{d} \mu(x)=e^{-u(x)}\, \mathrm{d} x$. Given a convex and compact set $K \subset \mathbb{R}^n$ of class then In particular, if $\mu$ satisfies eq:strong_dim_BM for every $K$ of class $C^2_+$, $\mu$ sati

Theorems & Definitions (4)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Corollary 2.2