Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm

TL;DR

This paper constructs special geodesics in Wasserstein space with bounded densities to derive geometric and functional inequalities from Sturm's curvature-dimension conditions in metric spaces.

Contribution

It introduces a method to build geodesics with bounded densities and links Ricci curvature bounds to Poincaré and measure contraction properties in metric measure spaces.

Findings

Derived local Poincaré inequality from Ricci bounds

Established measure contraction property under Sturm's conditions

Connected weak displacement convexity to Poincaré inequality

Abstract

We construct geodesics in the Wasserstein space of probability measure along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincar\'e inequality and the measure contraction property follow from the Ricci curvature bounds defined by Sturm. We also show for a large class of convex functionals that a local Poincar\'e inequality is implied by the weak displacement convexity of the functional.