Journal
JOURNAL OF FUNCTIONAL ANALYSIS
Volume 263, Issue 4, Pages 896-924Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jfa.2012.05.006
Keywords
Ricci curvature; Metric measure space; Poincare inequality; Measure contraction property
Categories
Funding
- European Project ERC AdG *GeMeThNES*
- Academy of Finland [137528]
- Academy of Finland (AKA) [137528, 137528] Funding Source: Academy of Finland (AKA)
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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 Poincare 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 Poincare inequality is implied by the weak displacement convexity of the functional. (C) 2012 Elsevier Inc. All rights reserved.
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