Let $X^n$ be an Alexandrov space, and $f: X^n\to \mathbb R^k$ a regular map, does the level set necessary be an Alexandrov space?


In my mind, the intrinsic metric on the level set is 'comparable' to the ambient metric, but is it necessary an Alexandrov space?



Is Level set of Regular functions in Alexandrov spaces again an Alex. space?
The answer is "no" even for regular semiconcave function $f:X\to\mathbb R$


If $f:X\to \mathbb R$ is convex then it is a long-standing open problem.


