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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
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[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
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