A note on the quantitative local version of the log-Brunn-Minkowski inequality

TL;DR

This paper establishes a local version of the log-Brunn-Minkowski inequality, proving it for a ball and symmetric convex bodies near it in a smooth neighborhood, advancing understanding of this geometric inequality.

Contribution

It provides a local proof of the log-Brunn-Minkowski inequality for specific convex bodies, a step towards its broader validation.

Findings

Proves the inequality for a ball and symmetric convex bodies near it

Uses a $C^2$ neighborhood approach

Advances the understanding of the log-Brunn-Minkowski inequality

Abstract

We prove that the log-Brunn-Minkowski inequality \begin{equation*} |\lambda K+_0 (1-\lambda)L|\geq |K|^{\lambda}|L|^{1-\lambda} \end{equation*} (where $|\cdot|$ is the Lebesgue measure and $+_0$ is the so-called log-addition) holds when $K\subset\mathbb{R}^n$ is a ball and $L$ is a symmetric convex body in a suitable $C^2$ neighborhood of $K$.