Dimensionality and the stability of the Brunn-Minkowski inequality

TL;DR

This paper establishes improved stability estimates for the Brunn-Minkowski inequality in convex geometry, with results that become stronger in higher dimensions and relate to the central limit theorem for convex sets.

Contribution

It provides novel stability estimates that improve with dimension, connecting geometric inequalities to probabilistic limit theorems.

Findings

Stability estimates for the Brunn-Minkowski inequality that improve as dimension increases

Equivalence of stability results to a thin shell bound in convex geometry

Application to the proof of the central limit theorem for convex sets

Abstract

We prove stability estimates for the Brunn-Minkowski inequality for convex sets. Unlike existing stability results, our estimates improve as the dimension grows. Our results are equivalent to a thin shell bound, which is one of the central ingredients in the proof of the central limit theorem for convex sets.