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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