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The main result of this article states that the (K;N)-cone over some metricmeasure space satisfies the reduced Riemannian curvature-dimension conditionRCD^*(KN;N+1) if and only if the underlying space satisfies RCD^*(N-1;N). Theproof uses a characterization of reduced Riemannian curvature-dimension boundsby Bochner's inequality that was established for metric measure spaces byErbar, Kuwada and Sturm and independently by Ambrosio, Mondino and Savar\'e. Ascorollary of our result and the Cheeger-Gigli-Gromoll splitting theorem weobtain a maximal diameter theorem for metric measure spaces that satisfy areduced Riemannian curvature-dimension condition.

Cones over metric measure spaces and the maximal diameter theorem