A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures

TL;DR

This paper establishes a near-optimal universal bound for the dimensional Brunn-Minkowski inequality in the context of log-concave measures, advancing the understanding of this fundamental geometric inequality.

Contribution

It provides a new universal bound with explicit dependence on dimension for the Brunn-Minkowski inequality for log-concave measures, improving previous results.

Findings

Proves a bound with exponent c_n ≥ n^{-4 - o(1)} for symmetric convex sets.

Progress towards the dimensional Brunn-Minkowski conjecture.

Bound improves for certain classes of log-concave measures.

Abstract

We show that for any log-concave measure $\mu$ on $\mathbb{R}^n$, any pair of symmetric convex sets $K$ and $L$, and any $\lambda\in [0,1],$ $$\mu((1-\lambda) K+\lambda L)^{c_n}\geq (1-\lambda) \mu(K)^{c_n}+\lambda\mu(L)^{c_n},$$ where $c_n\geq n^{-4-o(1)}.$ This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Gardner, Zvavitch \cite{GZ}, Colesanti, L, Marsiglietti \cite{CLM}). Moreover, our bound improves for various special classes of log-concave measures.