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.  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.   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, (2010), 88.   Google Scholar

[5]

D. Bevan, Sets of points determining only acute angles and some related colouring problems,, Electron. J. Combin., 13 (2006).   Google Scholar

[6]

L. V. Buchok, Two new approaches to obtaining estimates in the Danzer-Grünbaum problem,, Math. Notes, 87 (2010), 489.  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.  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.  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), 275.   Google Scholar

[10]

P. Erdős, Some unsolved problems,, Michigan Math. J., 4 (1957), 291.  doi: 10.1307/mmj/1028997963.  Google Scholar

[11]

K. Grove and P. Petersen, A radius sphere theorem,, Invent. Math., 112 (1993), 577.  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.  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.   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.   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, (1994), 517.   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.  doi: 10.1007/s000390050050.  Google Scholar

[24]

A. Petrunin, Semiconcave functions in Alexandrov's geometry,, in Surveys in Differential Geometry, (2007), 137.   Google Scholar

[25]

A. Wörner, Boundary Strata of Nonnegatively Curved Alexandrov Spaces and a Splitting Theorem,, Ph.D Thesis, (2010).   Google Scholar

show all references

References:
[1]

E. Ackerman and O. Ben-Zwib, On sets of points that determine only acute angles,, European J. Combin., 30 (2009), 908.  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.   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, (2010), 88.   Google Scholar

[5]

D. Bevan, Sets of points determining only acute angles and some related colouring problems,, Electron. J. Combin., 13 (2006).   Google Scholar

[6]

L. V. Buchok, Two new approaches to obtaining estimates in the Danzer-Grünbaum problem,, Math. Notes, 87 (2010), 489.  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.  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.  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), 275.   Google Scholar

[10]

P. Erdős, Some unsolved problems,, Michigan Math. J., 4 (1957), 291.  doi: 10.1307/mmj/1028997963.  Google Scholar

[11]

K. Grove and P. Petersen, A radius sphere theorem,, Invent. Math., 112 (1993), 577.  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.  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.   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.   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, (1994), 517.   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.  doi: 10.1007/s000390050050.  Google Scholar

[24]

A. Petrunin, Semiconcave functions in Alexandrov's geometry,, in Surveys in Differential Geometry, (2007), 137.   Google Scholar

[25]

A. Wörner, Boundary Strata of Nonnegatively Curved Alexandrov Spaces and a Splitting Theorem,, Ph.D Thesis, (2010).   Google Scholar

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