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

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 \({{\,\textrm{RCD}\,}}(K,N)\) space X has a noncollapsed \({{\,\textrm{RCD}\,}}(K,N)\) 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.