Abstract or Additional Information
We introduce a synthetic lower mean curvature bound for the topological boundary of a subset in a metric measure space that satisfies a lower Ricci curvature bound 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 a theorem about 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 subset is isometric to a geodesic ball centered at the tip of an Euclidean cone. This generalizes theorems in smooth context by Kasue and Sakurai to a singular framework. This is a joint work with Annegret Burtscher, Robert McCann and Eric Woolgar.