Local curvature-dimension condition implies measure-contraction property

We prove that for non-branching metric measure spaces the local curvature condition CDloc(K,N) implies the global version of MCP(K,N). The curvature condition CD(K,N) introduced by the second author and also studied by Lott and Villani is the generalization to metric measure space of lower bounds on Ricci curvature together with upper bounds on the dimension. This paper is the following step of Bacher and Sturm (2010) [1] where it is shown that CDloc(K,N) is equivalent to a global condition CD⁎(K,N), slightly weaker than the usual CD(K,N). It is worth pointing out that our result implies sharp Bishop–Gromov volume growth inequality and sharp Poincaré inequality.