# Existence of shortest paths in complete Alexandrov spaces

Let $X$ be complete finite dimensional Alexandrov space with curvature bounded from below. Is it true that any two points can be connected by a shortest path? If this is not true in general, it it true under some assumptions on $X$? (E.g. this is certainly true for $X$ being a complete smooth Riemannian manifold.)

• If $(X,d)$ is a complete, locally-compact path metric space, then any two points can be joined by a minimizing geodesic (using the Arzela-Ascoli and lowersemicontinuity of length). Finite-dimensional complete Alexandrov spaces are locally compact. – J. Martel Apr 11 '15 at 18:28
• @J.Martel: Do you need just local compactness or that any closed ball is compact? (in any case, you are right since complete finite dimensional Alexandrov spaces do have this property, as I learned.) – MKO Apr 11 '15 at 18:33