On the stability of Brunn-Minkowski type inequalities

TL;DR

This paper investigates the stability of the Log-Brunn-Minkowski inequality and related inequalities for symmetric convex sets, providing new stability results, infinitesimal versions, and implications for measures like Gaussian, with distinctions based on symmetry.

Contribution

It establishes stability results for Log-Brunn-Minkowski and dimensional inequalities near a ball, and derives infinitesimal versions and strong inequalities for convex sets.

Findings

Stability results for Log-Brunn-Minkowski near a ball

Infinitesimal version of the Log-Brunn-Minkowski inequality

Strong Poincaré-type inequality for convex sets

Abstract

Log-Brunn-Minkowski inequality was conjectured by Bor\"oczky, Lutwak, Yang and Zhang \cite{BLYZ}, and it states that a certain strengthening of the classical Brunn-Minkowski inequality is admissible in the case of symmetric convex sets. It was recently shown by Nayar, Zvavitch, the second and the third authors \cite{LMNZ}, that Log-Brunn-Minkowski inequality implies a certain dimensional Brunn-Minkowski inequality for log-concave measures, which in the case of Gaussian measure was conjectured by Gardner and Zvavitch \cite{GZ}. In this note, we obtain stability results for both Log-Brunn-Minkowski and dimensional Brunn-Minkowski inequalities for rotation invariant log-conave measures near a ball. Remarkably, the assumption of symmetry is only necessary for Log-Brunn-Minkowski stability, which emphasizes an important difference between the two conjectured inequalities. Also, we determine the infinitesimal version of the log-Brunn-Minkowski inequality. As a consequence, we obtain a strong Poincar\'{e}-type inequality in the case of unconditional convex sets, as well as for symmetric convex sets on the plane. Additionally, we derive an infinitesimal equivalent version of the B-conjecture for an arbitrary measure.