American Institute of Mathematical Sciences

2014, 21: 120-125. doi: 10.3934/era.2014.21.120

Number of extremal subsets in Alexandrov spaces and rigidity

 1 St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, 191023, St. Petersburg

Received  April 2014 Published  July 2014

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.
Citation: Nina Lebedeva. Number of extremal subsets in Alexandrov spaces and rigidity. Electronic Research Announcements, 2014, 21: 120-125. doi: 10.3934/era.2014.21.120
References:
 [1] E. Ackerman and O. Ben-Zwib, On sets of points that determine only acute angles, European J. Combin., 30 (2009), 908-910. doi: 10.1016/j.ejc.2008.07.020.  Google Scholar [2] S. Alexander and R. Bishop, A cone splitting theorem for Alexandrov spaces, Pacific Journal of Mathematics, 218 (2005), 1-16. Google Scholar [3] S. Alexander, V. Kapovitch and A. Petrunin, Alexandrov Geometry., Available from: , ().   Google Scholar [4] S. Alexander, V. Kapovitch and A. Petrunin, Alexandrov meets Kirszbraun, in Proceedings of the Gökova Geometry-Topology Conference 2010 (eds. S. Akbulut, D. Auroux and T. Onder), International Press, Somerville, MA, 2011, 88-109.  Google Scholar [5] D. Bevan, Sets of points determining only acute angles and some related colouring problems, Electron. J. Combin., 13 (2006), Research Paper 12, 24 pp. (electronic).  Google Scholar [6] L. V. Buchok, Two new approaches to obtaining estimates in the Danzer-Grünbaum problem, Math. Notes, 87 (2010), 489-496. doi: 10.1134/S0001434610030272.  Google Scholar [7] Yu. Burago, M. Gromov and G. Perel'man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk, 47 (1992), 3-51, 222; translation in Russian Math. Surveys, 47 (1992), 1-58. doi: 10.1070/RM1992v047n02ABEH000877.  Google Scholar [8] L. Danzer and B. Grünbaun, Über zwei Probleme bezüglich konvexer Körper von P. Erdős und von V. L. Klee, (German) Math. Z., 79 (1962), 95-99. doi: 10.1007/BF01193107.  Google Scholar [9] P. Erdős and Z. Fűredi, The greatest angle among n points in the d-dimensional Euclidean space, in Combinatorial Mathematics (Marseille-Luminy, 1981), North-Holland Math. Stud., 75, North-Holland, Amsterdam, 1983, 275-283.  Google Scholar [10] P. Erdős, Some unsolved problems, Michigan Math. J., 4 (1957), 291-300. doi: 10.1307/mmj/1028997963.  Google Scholar [11] K. Grove and P. Petersen, A radius sphere theorem, Invent. Math., 112 (1993), 577-583. doi: 10.1007/BF01232447.  Google Scholar [12] , V. Kapovich,, Private conversation., ().   Google Scholar [13] U. Lang and V. Shroeder, Kirszbraun's theorem and metric spaces of bounded curvature, Geom. Funct. Anal., 7 (1997), 535-560. doi: 10.1007/s000390050018.  Google Scholar [14] N. Lebedeva, Number of subgroups in a Bieberbach group., Available from: , ().   Google Scholar [15] N. Lebedeva and A. Petrunin, Local characterization of polyhedral spaces,, preprint, ().   Google Scholar [16] N. Li, Volume and gluing rigidity in Alexandrov geometry., Available from: , ().   Google Scholar [17] G. Ya. Perel'man, Elements of Morse theory on Aleksandrov spaces, St. Petersbg. Math. J., 5 (1994), 205-213.  Google Scholar [18] G. Ya. Perel'man and A. M. Petrunin, Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem, (Russian) Algebra i Analiz, 5 (1993), 242-256; translation in St. Petersburg Math. J., 5 (1994), 215-227.  Google Scholar [19] G. Ya. Perel'man and A. M. Petrunin, Quasigeodesics and gradient curves in Alexandrov spaces, preprint, 1994. Google Scholar [20] , G. Ya. Perel'man,, Private conversation., ().   Google Scholar [21] G. Ya. Perel'man, Spaces with curvature bounded below, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 517-525.  Google Scholar [22] , A. Petrunin,, Private conversation., ().   Google Scholar [23] A. Petrunin, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal., 8 (1998), 123-148. doi: 10.1007/s000390050050.  Google Scholar [24] A. Petrunin, Semiconcave functions in Alexandrov's geometry, in Surveys in Differential Geometry, Vol. XI, Int. Press, Somerville, MA, 2007, 137-201. Google Scholar [25] A. Wörner, Boundary Strata of Nonnegatively Curved Alexandrov Spaces and a Splitting Theorem, Ph.D Thesis, Westfälischen Wilhelms-Universität Münster, 2010. Available from: https://ivv5.uni-muenster.de/u/andreas.woerner/files/diss_woerner.pdf. Google Scholar

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References:
 [1] E. Ackerman and O. Ben-Zwib, On sets of points that determine only acute angles, European J. Combin., 30 (2009), 908-910. doi: 10.1016/j.ejc.2008.07.020.  Google Scholar [2] S. Alexander and R. Bishop, A cone splitting theorem for Alexandrov spaces, Pacific Journal of Mathematics, 218 (2005), 1-16. Google Scholar [3] S. Alexander, V. Kapovitch and A. Petrunin, Alexandrov Geometry., Available from: , ().   Google Scholar [4] S. Alexander, V. Kapovitch and A. Petrunin, Alexandrov meets Kirszbraun, in Proceedings of the Gökova Geometry-Topology Conference 2010 (eds. S. Akbulut, D. Auroux and T. Onder), International Press, Somerville, MA, 2011, 88-109.  Google Scholar [5] D. Bevan, Sets of points determining only acute angles and some related colouring problems, Electron. J. Combin., 13 (2006), Research Paper 12, 24 pp. (electronic).  Google Scholar [6] L. V. Buchok, Two new approaches to obtaining estimates in the Danzer-Grünbaum problem, Math. Notes, 87 (2010), 489-496. doi: 10.1134/S0001434610030272.  Google Scholar [7] Yu. Burago, M. Gromov and G. Perel'man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk, 47 (1992), 3-51, 222; translation in Russian Math. Surveys, 47 (1992), 1-58. doi: 10.1070/RM1992v047n02ABEH000877.  Google Scholar [8] L. Danzer and B. Grünbaun, Über zwei Probleme bezüglich konvexer Körper von P. Erdős und von V. L. Klee, (German) Math. Z., 79 (1962), 95-99. doi: 10.1007/BF01193107.  Google Scholar [9] P. Erdős and Z. Fűredi, The greatest angle among n points in the d-dimensional Euclidean space, in Combinatorial Mathematics (Marseille-Luminy, 1981), North-Holland Math. Stud., 75, North-Holland, Amsterdam, 1983, 275-283.  Google Scholar [10] P. Erdős, Some unsolved problems, Michigan Math. J., 4 (1957), 291-300. doi: 10.1307/mmj/1028997963.  Google Scholar [11] K. Grove and P. Petersen, A radius sphere theorem, Invent. Math., 112 (1993), 577-583. doi: 10.1007/BF01232447.  Google Scholar [12] , V. Kapovich,, Private conversation., ().   Google Scholar [13] U. Lang and V. Shroeder, Kirszbraun's theorem and metric spaces of bounded curvature, Geom. Funct. Anal., 7 (1997), 535-560. doi: 10.1007/s000390050018.  Google Scholar [14] N. Lebedeva, Number of subgroups in a Bieberbach group., Available from: , ().   Google Scholar [15] N. Lebedeva and A. Petrunin, Local characterization of polyhedral spaces,, preprint, ().   Google Scholar [16] N. Li, Volume and gluing rigidity in Alexandrov geometry., Available from: , ().   Google Scholar [17] G. Ya. Perel'man, Elements of Morse theory on Aleksandrov spaces, St. Petersbg. Math. J., 5 (1994), 205-213.  Google Scholar [18] G. Ya. Perel'man and A. M. Petrunin, Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem, (Russian) Algebra i Analiz, 5 (1993), 242-256; translation in St. Petersburg Math. J., 5 (1994), 215-227.  Google Scholar [19] G. Ya. Perel'man and A. M. Petrunin, Quasigeodesics and gradient curves in Alexandrov spaces, preprint, 1994. Google Scholar [20] , G. Ya. Perel'man,, Private conversation., ().   Google Scholar [21] G. Ya. Perel'man, Spaces with curvature bounded below, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 517-525.  Google Scholar [22] , A. Petrunin,, Private conversation., ().   Google Scholar [23] A. Petrunin, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal., 8 (1998), 123-148. doi: 10.1007/s000390050050.  Google Scholar [24] A. Petrunin, Semiconcave functions in Alexandrov's geometry, in Surveys in Differential Geometry, Vol. XI, Int. Press, Somerville, MA, 2007, 137-201. Google Scholar [25] A. Wörner, Boundary Strata of Nonnegatively Curved Alexandrov Spaces and a Splitting Theorem, Ph.D Thesis, Westfälischen Wilhelms-Universität Münster, 2010. Available from: https://ivv5.uni-muenster.de/u/andreas.woerner/files/diss_woerner.pdf. Google Scholar
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