Stability for the Brunn-Minkowski and Riesz rearrangement inequalities, with applications to Gaussian concentration and finite range non-local isoperimetry

TL;DR

This paper introduces a general method to enhance concentration inequalities by leveraging improvements in isoperimetric inequalities, leading to robust results for the Brunn-Minkowski, Gaussian concentration, and Riesz rearrangement inequalities, with applications in statistical mechanics.

Contribution

It presents a simple, general argument to improve concentration inequalities based on isoperimetric inequalities, and applies it to key inequalities with practical applications.

Findings

Robust improvements of the Brunn-Minkowski inequality for Minkowski sums.

Enhanced Gaussian concentration inequality results.

Application to finite-range nonlocal isoperimetric problems in statistical mechanics.

Abstract

We provide a simple, general argument to obtain improvements of concentration-type inequalities starting from improvements of their corresponding isoperimetric-type inequalities. We apply this argument to obtain robust improvements of the Brunn-Minkowski inequality (for Minkowski sums between generic sets and convex sets) and of the Gaussian concentration inequality. The former inequality is then used to obtain a robust improvement of the Riesz rearrangement inequality under certain natural conditions. These conditions are compatible with the applications to a finite-range nonlocal isoperimetric problem arising in statistical mechanics.