American Institute of Mathematical Sciences

Advanced
January  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

[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

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

[1]

Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics & Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005
[2]

Yuri Kalinin, Volker Reitmann, Nayil Yumaguzin. Asymptotic behavior of Maxwell's equation in one-space dimension with thermal effect. Conference Publications, 2011, 2011 (Special) : 754-762. doi: 10.3934/proc.2011.2011.754
[3]

Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155
[4]

M. Petcu. Euler equation in a channel in space dimension 2 and 3. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 755-778. doi: 10.3934/dcds.2005.13.755
[5]

Eric L. Grinberg, Haizhong Li. The Gauss-Bonnet-Grotemeyer Theorem in space forms. Inverse Problems & Imaging, 2010, 4 (4) : 655-664. doi: 10.3934/ipi.2010.4.655
[6]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511
[7]

Luciano Pandolfi. Riesz systems and moment method in the study of viscoelasticity in one space dimension. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1487-1510. doi: 10.3934/dcdsb.2010.14.1487
[8]

Kyouhei Wakasa. The lifespan of solutions to semilinear damped wave equations in one space dimension. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1265-1283. doi: 10.3934/cpaa.2016.15.1265
[9]

Igor Kukavica. On Fourier parametrization of global attractors for equations in one space dimension. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 553-560. doi: 10.3934/dcds.2005.13.553
[10]

Marco Di Francesco, Simone Fagioli, Massimiliano Daniele Rosini, Giovanni Russo. Deterministic particle approximation of the Hughes model in one space dimension. Kinetic & Related Models, 2017, 10 (1) : 215-237. doi: 10.3934/krm.2017009
[11]

Raffaele Chiappinelli. Eigenvalues of homogeneous gradient mappings in Hilbert space and the Birkoff-Kellogg theorem. Conference Publications, 2007, 2007 (Special) : 260-268. doi: 10.3934/proc.2007.2007.260
[12]

Marcello D'Abbicco, Sandra Lucente. NLWE with a special scale invariant damping in odd space dimension. Conference Publications, 2015, 2015 (special) : 312-319. doi: 10.3934/proc.2015.0312
[13]

Chiu-Ya Lan, Huey-Er Lin, Shih-Hsien Yu. The Green's functions for the Broadwell Model in a half space problem. Networks & Heterogeneous Media, 2006, 1 (1) : 167-183. doi: 10.3934/nhm.2006.1.167
[14]

Elio E. Espejo, Masaki Kurokiba, Takashi Suzuki. Blowup threshold and collapse mass separation for a drift-diffusion system in space-dimension two. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2627-2644. doi: 10.3934/cpaa.2013.12.2627
[15]

Yuji Sagawa, Hideaki Sunagawa. The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5743-5761. doi: 10.3934/dcds.2016052
[16]

Fred C. Pinto. Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 117-136. doi: 10.3934/dcds.1999.5.117
[17]

Laurent Gosse. Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension. Kinetic & Related Models, 2012, 5 (2) : 283-323. doi: 10.3934/krm.2012.5.283
[18]

Jiahang Che, Li Chen, Simone GÖttlich, Anamika Pandey, Jing Wang. Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1013-1036. doi: 10.3934/cpaa.2017049
[19]

Debanjana Mitra, Mythily Ramaswamy, Jean-Pierre Raymond. Largest space for the stabilizability of the linearized compressible Navier-Stokes system in one dimension. Mathematical Control & Related Fields, 2015, 5 (2) : 259-290. doi: 10.3934/mcrf.2015.5.259
[20]

Kei Nakamura, Tohru Ozawa. Finite charge solutions to cubic Schrödinger equations with a nonlocal nonlinearity in one space dimension. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 789-801. doi: 10.3934/dcds.2013.33.789

2018 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences