Mass transportation and rough curvature bounds for discrete spaces |
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Authors: | Anca-Iuliana Bonciocat |
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Affiliation: | a Institute of Mathematics “S. Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania b Institute for Applied Mathematics, University of Bonn, Poppelsdorfer Allee 82/1, 53115 Bonn, Germany |
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Abstract: | We introduce and study rough (approximate) lower curvature bounds for discrete spaces and for graphs. This notion agrees with the one introduced in [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press] and [K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65-131], in the sense that the metric measure space which is approximated by a sequence of discrete spaces with rough curvature ?K will have curvature ?K in the sense of [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press; K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65-131]. Moreover, in the converse direction, discretizations of metric measure spaces with curvature ?K will have rough curvature ?K. We apply our results to concrete examples of homogeneous planar graphs. |
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Keywords: | Optimal transport Ricci curvature GH-limits Graphs Concentration of measure |
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