Let $X$ be locally compact Alexandrov space whose curvature satisfies both inequalities $\geq K$ and $\leq K$. What can be said about such a space? Is it locally isometric to the standard Riemannian manifold of constant curvature $K$ (e.g. sphere, Euclidean, or hyperbolic space)?
(Recall that if $X$ is a smooth Riemannian manifold then the answer is positive.)