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 $K$?**



Is the boundary of Alexandrov space again an Alexandrov space?
This is an open problem. 


It is a special case of the following question: 



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> Is it true that every extremal subset is again an Alexandrov space?
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The answer to this question is "No". Petrunin has constructed a counterexample in codimension three, [here](http://library.msri.org/books/Book30/files/petrunin.pdf).


