Calculus of Variations and Geometric Measure Theory

T. Rajala

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

created by rajala1 on 24 Nov 2011
modified on 14 Nov 2012


Published Paper

Inserted: 24 nov 2011
Last Updated: 14 nov 2012

Journal: J. Funct. Anal.
Volume: 263
Number: 4
Pages: 896-924
Year: 2012


We construct geodesics in the Wasserstein space of probability measures 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 Poincaré 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 Poincaré inequality is implied by the weak displacement convexity of the functional.

Tags: GeMeThNES
Keywords: Ricci curvature, Poincare inequality, Geodesics, metric measure spaces, measure contraction property