TITLE: Is the boundary of Alexandrov space again an Alexandrov space?
QUESTION [5 upvotes]: 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 $K$?

REPLY [5 votes]: 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.