A curved Brunn--Minkowski inequality on the discrete hypercube

2010/11/22 by Ollivier, Yann, Villani, Cédric · 2 citations
#51K10 #52A40 #52C99 #53C23 #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.1011.4779

Abstract

We compare two approaches to Ricci curvature on non-smooth spaces, in the case of the discrete hypercube \0,1\^N. While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement convexity property of Lott, Sturm and the second author could not be fully implemented. Yet along the way we get new results of a combinatorial and probabilistic nature, including a curved Brunn--Minkowski inequality on the discrete hypercube.

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• From Brunn-Minkowski to Prékopa-Leindler and Borell-Brascamp-Lieb: discrete inequalities
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