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,, \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]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

[2]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[3]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[4]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037

[5]

Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230

[6]

Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048

[7]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

[8]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

[9]

Azmy S. Ackleh, Nicolas Saintier. Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1469-1497. doi: 10.3934/dcdsb.2020169

[10]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[11]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390

[12]

Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013

[13]

Ali Mahmoodirad, Harish Garg, Sadegh Niroomand. Solving fuzzy linear fractional set covering problem by a goal programming based solution approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020162

[14]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395

[15]

Vadim Azhmyakov, Juan P. Fernández-Gutiérrez, Erik I. Verriest, Stefan W. Pickl. A separation based optimization approach to Dynamic Maximal Covering Location Problems with switched structure. Journal of Industrial & Management Optimization, 2021, 17 (2) : 669-686. doi: 10.3934/jimo.2019128

[16]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[17]

Lisa Hernandez Lucas. Properties of sets of Subspaces with Constant Intersection Dimension. Advances in Mathematics of Communications, 2021, 15 (1) : 191-206. doi: 10.3934/amc.2020052

[18]

Hong Fu, Mingwu Liu, Bo Chen. Supplier's investment in manufacturer's quality improvement with equity holding. Journal of Industrial & Management Optimization, 2021, 17 (2) : 649-668. doi: 10.3934/jimo.2019127

[19]

Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015

[20]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

2019 Impact Factor: 0.5

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]