# Cones over metric measure spaces and a maximal diameter theorem

##
Christian Ketterer

created by paolini on 25 Apr 2014

modified by gigli on 25 May 2014

**Abstract.**

In this talk I will present a maximal diameter theorem for metric
measure spaces that satisfy the reduced Riemannian curvature-dimension
condition $RCD^*(N-1,N)$. That is, if there is equality in the
Bonnet-Myers diameter estimate, the space is a spherical suspension.
The proof is an application of the Gigli-Cheeger-Gromoll splitting
theorem, and the stability of Riemannian curvature-dimension bounds
under coning. More precisely, the latter 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 $RCD^*(N-1,N)$. I will describe the main
ideas of the proof.