On the intrinsic and extrinsic boundary for metric measure spaces with lower curvature bounds

Abstract

We show that if an Alexandrov space $X$ has an Alexandrov subspace $\bar{\Omega }$ of the same dimension disjoint from the boundary of $X$, then the topological boundary of $\bar{\Omega }$ coincides with its Alexandrov boundary. Similarly, if a noncollapsed RCD(K,N) space $X$ has a noncollapsed RCD(K,N) subspace $\bar{\Omega }$ disjoint from boundary of $X$ and with mild boundary condition, then the topological boundary of $\bar{\Omega }$ coincides with its De Philippis-Gigli boundary. We then discuss some consequences about convexity of such type of equivalence.

19 pages