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A curved Brunn-Minkowski inequality on the discrete hypercube

Yann Ollivier 1, 2, * Cédric Villani 3
* Corresponding author
2 TAO - Machine Learning and Optimisation
CNRS - Centre National de la Recherche Scientifique : UMR8623, Inria Saclay - Ile de France, UP11 - Université Paris-Sud - Paris 11, LRI - Laboratoire de Recherche en Informatique
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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Submitted on : Wednesday, September 4, 2013 - 2:10:14 PM
Last modification on : Thursday, July 8, 2021 - 3:46:04 AM

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Yann Ollivier, Cédric Villani. A curved Brunn-Minkowski inequality on the discrete hypercube. SIAM Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2012, 26 (3), pp.983-996. ⟨10.1137/11085966X⟩. ⟨inria-00540479⟩

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