Stability for Borell-Brascamp-Lieb inequalities

TL;DR

This paper investigates the stability of Borell-Brascamp-Lieb inequalities, showing that near equality implies functions are close to p-concave and similar up to homothety.

Contribution

It provides new stability results linking near equality in these inequalities to geometric and functional closeness of the involved functions.

Findings

Functions are close to p-concave when near equality holds.

Functions nearly coincide up to homothety in the near-equality case.

Stability results quantify the closeness of functions in the inequality context.

Abstract

We study stability issues for the so-called Borell-Brascamp-Lieb inequalities, proving that when near equality is realized, the involved functions must be $L^1$-close to be $p$-concave and to coincide up to homotheties of their graphs.