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

On Helly's theorem in geodesic spaces

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.
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, Russian, Fundam. Prikl. Mat., 8 (2002), 365-405.

[2]

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

[3]

D. Burago and S. Ivanov, Polyhedral Finsler spaces with locally unique geodesics, Adv. Math., 247 (2013), 343-355. doi: 10.1016/j.aim.2013.07.007.

[4]

H. Busemann, Spaces with non-positive curvature, Acta Math., 80 (1948), 259-310. doi: 10.1007/BF02393651.

[5]

H. E. Debrunner, Helly type theorems derived from basic singular homology, Amer. Math. Monthly, 77 (1970), 375-380. doi: 10.2307/2316144.

[6]

B. Farb, Group actions and Helly's theorem, Adv. Math., 222 (2009), 1574-1588. doi: 10.1016/j.aim.2009.06.004.

[7]

E. Helly, Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten, Monatsh. Math. Phys., 37 (1930), 281-302. doi: 10.1007/BF01696777.

[8]

R. N. Karasev, A topological central point theorem, Topology Appl., 159 (2012), 864-868. doi: 10.1016/j.topol.2011.12.002.

[9]

B. Kleiner, The local structure of length spaces with curvature bounded above, Math. Z., 231 (1999), 409-456. doi: 10.1007/PL00004738.

[10]

B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für $n$-dimensionale Simplexe, Fund. Math., 14 (1929), 132-137.

[11]

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

[12]

L. Montejano, A new topological Helly theorem, preprint, (2013). Available from: https://www.researchgate.net/publication/235626408.

show all references

References:
[1]

S. A. Bogatyĭ, The topological Helly theorem, Russian, Fundam. Prikl. Mat., 8 (2002), 365-405.

[2]

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

[3]

D. Burago and S. Ivanov, Polyhedral Finsler spaces with locally unique geodesics, Adv. Math., 247 (2013), 343-355. doi: 10.1016/j.aim.2013.07.007.

[4]

H. Busemann, Spaces with non-positive curvature, Acta Math., 80 (1948), 259-310. doi: 10.1007/BF02393651.

[5]

H. E. Debrunner, Helly type theorems derived from basic singular homology, Amer. Math. Monthly, 77 (1970), 375-380. doi: 10.2307/2316144.

[6]

B. Farb, Group actions and Helly's theorem, Adv. Math., 222 (2009), 1574-1588. doi: 10.1016/j.aim.2009.06.004.

[7]

E. Helly, Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten, Monatsh. Math. Phys., 37 (1930), 281-302. doi: 10.1007/BF01696777.

[8]

R. N. Karasev, A topological central point theorem, Topology Appl., 159 (2012), 864-868. doi: 10.1016/j.topol.2011.12.002.

[9]

B. Kleiner, The local structure of length spaces with curvature bounded above, Math. Z., 231 (1999), 409-456. doi: 10.1007/PL00004738.

[10]

B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für $n$-dimensionale Simplexe, Fund. Math., 14 (1929), 132-137.

[11]

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

[12]

L. Montejano, A new topological Helly theorem, preprint, (2013). Available from: https://www.researchgate.net/publication/235626408.

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