On Prekopa-Leindler inequalities on metric-measure spaces

TL;DR

This paper explores how Prekopa-Leindler and Borell-Brascamp-Lieb inequalities characterize Ricci curvature bounds and other geometric properties in metric-measure spaces, extending classical Riemannian results to more general spaces.

Contribution

It establishes that these functional inequalities characterize lower Ricci curvature bounds in measured length spaces, extending prior Riemannian results to a broader setting.

Findings

Prekopa-Leindler inequalities characterize Ricci curvature lower bounds.

Stability properties of inequalities in measured length spaces.

Connections between functional inequalities and geometric estimates.

Abstract

This work is devoted to the geometric analysis of metric-measure spaces satisfying a Prekopa-Leindler or a more general Borell-Brascamp-Lieb inequality. Completing the early investigations by Cordero-Erausquin, McCann and Schmuckenschlager, we show that these functional inequalities characterize lower bounds on the Ricci curvature on a Riemannian manifold, providing thus an alternate version of Ricci curvature lower bounds in measured length spaces to the recent developments by Lott, Villani and Sturm. We also investigate stability properties and geometric and functional inequalities, such as logarithmic Sobolev inequality and Bishop-Gromov diameter estimate, in measured length spaces satisfying a Prekopa-Leindler or a Borell-Brascamp-Lieb inequality.