Improved stability versions of the Pr\'ekopa-Leindler inequality

TL;DR

This paper establishes a dimension- and parameter-uniform stability estimate for the Prékopa-Leindler inequality using transport maps and trace inequalities, with applications to log-concave functions.

Contribution

It introduces a new uniform stability exponent for the Prékopa-Leindler inequality that is independent of dimension and the log-concavity parameter.

Findings

Derived a uniform stability exponent for the inequality

Proved a sharp stability result for log-concave functions in dimension 1

Extended stability results to radial functions in higher dimensions

Abstract

We consider the problem of stability for the Pr\'ekopa-Leindler inequality. Exploiting properties of the transport map between radially decreasing functions and a suitable functional version of the trace inequality, we obtain a uniform stability exponent for the Pr\'ekopa-Leindler inequality. Our result yields an exponent not only uniform in the dimension but also in the log-concavity parameter $\tau = \min(\lambda,1-\lambda)$ associated with its respective version of the Pr\'ekopa-Leindler inequality. As a further application of our methods, we prove a sharp stability result for log-concave functions in dimension 1, which also extends to a sharp stability result for log-concave radial functions in higher dimensions.