Cones over metric measure spaces and the maximal diameter theorem

The main result of this article states that the (K, N)-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD* (KN, N + 1) if and only if the underlying space satisfies RCD* (N - 1, N). The proof uses a characterization of reduced Riemannian curvature-dimension bounds by Bochner's inequality that was established for general metric measure spaces by Erbar, Kuwada and Sturm in [21] (independently, the same result has been announced by Ambrosio, Mondino and Savare). As a corollary of our result and the Gigli-Cheeger-Gromoll splitting theorem [25] we obtain a maximal diameter theorem in the context of metric measure spaces that satisfy the condition RCD* (C) 2014 Elsevier Masson SAS. All rights reserved.