On Borell-Brascamp-Lieb Inequalities on Metric Measure Spaces

In this paper we introduce the notion of a Borell-Brascamp-Lieb inequality for metric measure spaces (M,d,m) denoted by BBL(K,N) for two numbers K,N aaEuro parts per thousand a"e with N a parts per thousand yenaEuro parts per thousand 1. In the first part we prove that BBL(K,N) holds true on metric measure spaces satisfying a curvature-dimension condition CD(K,N) developed and studied by Lott and Villani in (Ann Math 169:903-991, 2007) as well as by Sturm in (Acta Math 196(1):133-177, 2006). The aim of the second part is to show that BBL(K,N) is stable under convergence of metric measure spaces with respect to the L (2)-transportation distance.