Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


Weak topologies in complete $ CAT(0)$ metric spaces

Author: Bijan Ahmadi Kakavandi
Journal: Proc. Amer. Math. Soc. 141 (2013), 1029-1039
MSC (2010): Primary 53C23; Secondary 54A10
Published electronically: July 27, 2012
MathSciNet review: 3003694
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider some open questions concerning $ \Delta $-convergence in complete $ CAT(0)$ metric spaces (i.e. Hadamard spaces). Suppose $ (X,d)$ is a Hadamard space such that the sets $ \{z \in X \vert \, d(x,z) \leq d(z,y) \}$ are convex for each $ x,y \in X$. We introduce a so-called half-space topology such that convergence in this topology is equivalent to $ \Delta $-convergence for any sequence in $ X$. For a major class of Hadamard spaces, our results answer positively open questions nos. 1, 2 and 3 in [W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008) 3689-3696]. Moreover, we give a new characterization of $ \Delta $-convergence and a new topology that we call the weak topology via a concept of a dual metric space. The relations between these topologies and the topology which is induced by the distance function have been studied. The paper concludes with some examples.

References [Enhancements On Off] (What's this?)

  • 1. B. Ahmadi Kakavandi, M. Amini, Duality and Subdifferential for Convex Functions on Complete $ CAT(0)$ Metric Spaces, Nonlinear Anal. 73 (2010) 3450-3455. MR 2680038 (2012b:47131)
  • 2. I.D. Berg, I.G. Nikolaev, Quasilinearization and curvature of Alexandrov spaces, Geom. Dedicata 133 (2008) 195-218. MR 2390077 (2008m:53167)
  • 3. M. Bridson, A. Haefliger, Metric Spaces of Nonpositive Curvature, Grundlehren Math. Wiss. 319, Springer-Verlag, Berlin-Heidelberg-New York (1999). MR 1744486 (2000k:53038)
  • 4. D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Graduate Studies in Math., 33, Amer. Math. Soc., Providence, RI (2001). MR 1835418 (2002e:53053)
  • 5. R. Espínola, A. Fernández-León, $ CAT(\kappa )$-spaces, weak convergence and fixed points, J. Math. Anal. Appl. 353 (2009) 410-427. MR 2508878 (2010d:47092)
  • 6. M. Gromov, S.M. Bates, Metric structures for Riemannian and non-Riemannian spaces, with appendices by M. Katz, P. Pansu and S. Semmes, ed. by J. Lafontaine and P. Pansu, Progr. Math. 152, Birkhäuser, Boston (1999). MR 1699320 (2000d:53065)
  • 7. J. Jöst, Nonpositive curvature: Geometric and analytic aspects, Lectures Math. ETH Zürich, Birkhäuser, Basel (1997). MR 1451625 (98g:53070)
  • 8. W.A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008) 3689-3696. MR 2416076 (2009m:54061)
  • 9. T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179-182. MR 0423139 (54:11120)
  • 10. Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967) 591-597. MR 0211301 (35:2183)
  • 11. E.N. Sosov, On analogues of weak convergence in a special metric space, Izv. Vyssh. Uchebn. Zaved. Mat. 5 (2004) 84-89; translation in Russian Math. (Iz. VUZ) 48 (5) (2004) 79-83. MR 2101682 (2005f:54008)
  • 12. K.T. Sturm, Probability Measures on Metric Spaces of Non-positive Curvature, Heat Kernels and Analysis on Manifolds, Graphs and Metric Spaces (Paris, 2002), 357-390, Contemp. Math. 338, Amer. Math. Soc., Providence, RI, 2003. MR 2039961 (2005d:60009)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C23, 54A10

Retrieve articles in all journals with MSC (2010): 53C23, 54A10

Additional Information

Bijan Ahmadi Kakavandi
Affiliation: Department of Mathematics, Shahid Beheshti University G. C., P.O. Box 1983963113, Tehran, Iran
Email: b{\textunderscore}

Keywords: $CAT(0)$-space, $Δ$-convergence, $w$-convergence, half-space topology, weak topology
Received by editor(s): July 31, 2011
Published electronically: July 27, 2012
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society