Let $X$ be a finite dimensional (possibly compact) Alexandrov space with curvature $\geq K$. Is it true that its boundary is again Alexandrov space with curvature bounded from below? If yes, is the curvature at least $K$?
This is an open problem.
It is a special case of the following question:
Is it true that every extremal subset is again an Alexandrov space?
The answer to this question is "No". Petrunin has constructed a counterexample in codimension three, here.