One dimensional weighted Ricci curvature and displacement convexity of entropies

TL;DR

This paper establishes that a lower bound on 1-weighted Ricci curvature is equivalent to entropy convexity in Wasserstein space, leading to interpolation inequalities like Prékopa-Leindler and Brunn-Minkowski.

Contribution

It provides a novel characterization linking weighted Ricci curvature bounds to entropy convexity and derives key inequalities from this relationship.

Findings

Equivalence between 1-weighted Ricci curvature lower bounds and entropy convexity.

Derivation of Prékopa-Leindler, Brascamp-Lieb, and Brunn-Minkowski inequalities under curvature bounds.

New insights into geometric analysis and optimal transport theory.

Abstract

In the present paper, we prove that a lower bound on the $1$-weighted Ricci curvature is equivalent to a convexity of entropies on the Wasserstein space. Based on such characterization, we provide some interpolation inequalities such as the Pr'ekopa-Leindler inequality, the Borel-Branscamp-Lieb inequality, and the Brunn-Minkowski inequality under the curvature bound.