Number of extremal subsets in Alexandrov spaces and rigidity

Abstract

In this paper we announce the following result. We show that any $n$-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of $\mathbb{R}^n$ by an action of a crystallographic group. We describe all such actions. We start with a history, results and open questions concerning estimates on the number of extremal subsets in Alexandrov spaces. Then we give the plan of the proof of our result; the complete proof will published elsewhere.