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Axioms 2014, 3(1), 119-139; doi:10.3390/axioms3010119

Ricci Curvature on Polyhedral Surfaces via Optimal Transportation

1 ENS de Lyon, 15 parvis René Descartes - BP 7000 69342 Lyon Cedex 07, France 2 Université Paris-Est, LAMA (UMR 8050), Marne-la-Vallée F-77454, France
* Author to whom correspondence should be addressed.
Received: 27 January 2014 / Revised: 17 February 2014 / Accepted: 18 February 2014 / Published: 6 March 2014
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The problem of correctly defining geometric objects, such as the curvature, is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor, with the classical Ricci curvature, when the space is a smooth manifold. Lin, Lu and Yau and Jost and Liu have used and extended this notion for graphs, giving estimates for the curvature and, hence, the diameter, in terms of the combinatorics. In this paper, we describe a method for computing the coarse Ricci curvature and give sharper results, in the specific, but crucial case of polyhedral surfaces.
Keywords: discrete curvature; optimal transportation; graph theory; discrete Laplacian; tiling discrete curvature; optimal transportation; graph theory; discrete Laplacian; tiling
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Loisel, B.; Romon, P. Ricci Curvature on Polyhedral Surfaces via Optimal Transportation. Axioms 2014, 3, 119-139.

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