Let $M$ be a smooth Riemannian manifold (without boundary). Let $X\subset M$ be a smooth compact submanifold with boundary, $\dim X=\dim M$.
Under what conditions $X$, equipped with the induced intrinsic metric, is an Alexandrov space with curvature bounded below?
A simple sufficient condition is some convexity of $X$: any point of $X$ has a neighborhood such that any two points of $X$ in it are connected (in $M$) by a unique shortest geodesic, and this geodesic is contained in $X$. Is this condition also necessary?