### Inscribed radius bounds for metric measure spaces with mean-H-convex boundary

Analysis and Geometry Seminar

25th March 2021, 3:15 pm – 4:15 pm

Online, Zoom (contact organisers for details)

I present a synthetic lower mean curvature bound for the topological boundary of a subset in a metric measure space that has Ricci curvature bounded from below in the sense of Lott, Sturm and Villani. This lower mean curvature bound coincides with the classical notion in smooth context. As application, I present sharp comparison estimates for the inscribed radius of such subsets. Moreover, in the context of RCD(0,N) metric measure spaces (Riemannian curvature-dimension condition) equality holds if and only if the intrinsic geometry of the subset is isometric to a geodesic ball centred at the tip of a Euclidean cone, generalizing theorems by Kasue and Sakurai in smooth context. This is a joint work with Annegret Burtscher, Robert McCann and Eric Woolgar.

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