## Alexandrov spaces with maximal number of extremal points.(English)Zbl 1318.53074

Summary: 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.

### MSC:

 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov) 20F55 Reflection and Coxeter groups (group-theoretic aspects) 51K10 Synthetic differential geometry
Full Text:

### References:

 [1] S B Alexander, R L Bishop, A cone splitting theorem for Alexandrov spaces, Pacific J. Math. 218 (2005) 1 · Zbl 1125.53023 [2] Y Burago, M Gromov, G Perelman, A D Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992) 3, 222 · Zbl 0802.53018 [3] L Danzer, B Grünbaum, Über zwei Probleme bezüglich konvexer Körper von P Erd\Hos und von V L Klee, Math. Z. 79 (1962) 95 · Zbl 0188.27602 [4] P Erd\Hos, Some unsolved problems, Michigan Math. J. 4 (1957) 291 [5] V Kapovich, Private communication [6] N Lebedeva, Number of subgroups in a Bieberbach group [7] N Lebedeva, A Petrunin, Local characterization of polytopes, · Zbl 1333.52018 [8] N Li, Volume and gluing rigidity in Alexandrov geometry [9] G Perelman, Private communication [10] G Perelman, Spaces with curvature bounded below (editor S D Chatterji), Birkhäuser (1995) 517 · Zbl 0838.53033 [11] G Y Perelman, A M Petrunin, Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem, Algebra i Analiz 5 (1993) 242 · Zbl 0802.53019 [12] A Petrunin, Private communication
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.