American Institute of Mathematical Sciences

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  2014, 21: 109-112. doi: 10.3934/era.2014.21.109

On Helly's theorem in geodesic spaces

Sergei Ivanov 1,

1. 

St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russian Federation

Received  April 2014 Published  June 2014

In this note we show that Helly's Intersection Theorem holds for convex sets in uniquely geodesic spaces (in particular, in CAT(0) spaces) without the assumption that the convex sets are open or closed.
Keywords: Helly's Theorem, covering dimension, uniquely geodesic space, CAT(0) space..
Mathematics Subject Classification: 53C23, 52A35, 54F4.
Citation: Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109
References:
[1]

S. A. Bogatyĭ, The topological Helly theorem,, \emph{Russian, 8 (2002), 365.   Google Scholar

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M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319 (1999).  doi: 10.1007/978-3-662-12494-9.  Google Scholar

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H. Busemann, Spaces with non-positive curvature,, \emph{Acta Math.}, 80 (1948), 259.  doi: 10.1007/BF02393651.  Google Scholar

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H. E. Debrunner, Helly type theorems derived from basic singular homology,, \emph{Amer. Math. Monthly}, 77 (1970), 375.  doi: 10.2307/2316144.  Google Scholar

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B. Farb, Group actions and Helly's theorem,, \emph{Adv. Math.}, 222 (2009), 1574.  doi: 10.1016/j.aim.2009.06.004.  Google Scholar

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E. Helly, Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten,, \emph{Monatsh. Math. Phys.}, 37 (1930), 281.  doi: 10.1007/BF01696777.  Google Scholar

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R. N. Karasev, A topological central point theorem,, \emph{Topology Appl.}, 159 (2012), 864.  doi: 10.1016/j.topol.2011.12.002.  Google Scholar

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B. Kleiner, The local structure of length spaces with curvature bounded above,, \emph{Math. Z.}, 231 (1999), 409.  doi: 10.1007/PL00004738.  Google Scholar

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B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für $n$-dimensionale Simplexe,, \emph{Fund. Math.}, 14 (1929), 132.   Google Scholar

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Tverberg's theorem in CAT(0) spaces, Misha, http://mathoverflow.net/users/21684,, MathOverflow, (): 2013.   Google Scholar

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L. Montejano, A new topological Helly theorem,, preprint, (2013).   Google Scholar

show all references

References:
[1]

S. A. Bogatyĭ, The topological Helly theorem,, \emph{Russian, 8 (2002), 365.   Google Scholar

[2]

M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319 (1999).  doi: 10.1007/978-3-662-12494-9.  Google Scholar

[3]

D. Burago and S. Ivanov, Polyhedral Finsler spaces with locally unique geodesics,, \emph{Adv. Math.}, 247 (2013), 343.  doi: 10.1016/j.aim.2013.07.007.  Google Scholar

[4]

H. Busemann, Spaces with non-positive curvature,, \emph{Acta Math.}, 80 (1948), 259.  doi: 10.1007/BF02393651.  Google Scholar

[5]

H. E. Debrunner, Helly type theorems derived from basic singular homology,, \emph{Amer. Math. Monthly}, 77 (1970), 375.  doi: 10.2307/2316144.  Google Scholar

[6]

B. Farb, Group actions and Helly's theorem,, \emph{Adv. Math.}, 222 (2009), 1574.  doi: 10.1016/j.aim.2009.06.004.  Google Scholar

[7]

E. Helly, Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten,, \emph{Monatsh. Math. Phys.}, 37 (1930), 281.  doi: 10.1007/BF01696777.  Google Scholar

[8]

R. N. Karasev, A topological central point theorem,, \emph{Topology Appl.}, 159 (2012), 864.  doi: 10.1016/j.topol.2011.12.002.  Google Scholar

[9]

B. Kleiner, The local structure of length spaces with curvature bounded above,, \emph{Math. Z.}, 231 (1999), 409.  doi: 10.1007/PL00004738.  Google Scholar

[10]

B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für $n$-dimensionale Simplexe,, \emph{Fund. Math.}, 14 (1929), 132.   Google Scholar

[11]

Tverberg's theorem in CAT(0) spaces, Misha, http://mathoverflow.net/users/21684,, MathOverflow, (): 2013.   Google Scholar

[12]

L. Montejano, A new topological Helly theorem,, preprint, (2013).   Google Scholar

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