Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel

CiteScore 2017: 0.65

SCImago Journal Rank (SJR) 2017: 1.063
Source Normalized Impact per Paper (SNIP) 2017: 0.833

Mathematical Citation Quotient (MCQ) 2017: 0.86

Open Access
See all formats and pricing
More options …

Uniformly Convex Metric Spaces

Martin Kell
Published Online: 2014-12-10 | DOI: https://doi.org/10.2478/agms-2014-0015


In this paper the theory of uniformly convex metric spaces is developed. These spaces exhibit a generalized convexity of the metric from a fixed point. Using a (nearly) uniform convexity property a simple proof of reflexivity is presented and a weak topology of such spaces is analyzed. This topology, called coconvex topology, agrees with the usually weak topology in Banach spaces. An example of a CAT(0)-space with weak topology which is not Hausdorff is given. In the end existence and uniqueness of generalized barycenters is shown, an application to isometric group actions is given and a Banach-Saks property is proved.

Keywords: convex metric spaces; weak topologies; generalized barycenters; Banach-Saks property;


  • [1] M. Bacák, Convex analysis and optimization in Hadamard spaces, Walter de Gruyter & Co., Berlin, 2014. Google Scholar

  • [2] M. Bacák, B. Hua, J. Jost, M. Kell, and A. Schikorra, A notion of nonpositive curvature for general metric spaces, Differential Geometry and its Applications 38 (2015), 22–32. Google Scholar

  • [3] J. B. Baillon, Nonexpansive mappings and hyperconvex spaces, Contemp. Math (1988). Google Scholar

  • [4] M. R. Bridson and A. Häfliger, Metric Spaces of Non-Positive Curvature, Springer, 1999. CrossrefGoogle Scholar

  • [5] H. Busemann and B. B. Phadke,Minkowskian geometry, convexity conditions and the parallel axiom, Journal of Geometry 12 (1979), no. 1, 17–33. Google Scholar

  • [6] Th. Champion, L. De Pascale, and P. Juutinen, The1-Wasserstein Distance: Local Solutions and Existence of Optimal Transport Maps, SIAM Journal on Mathematical Analysis 40 (2008), no. 1, 1–20 (en). CrossrefWeb of ScienceGoogle Scholar

  • [7] J. A. Clarkson, Uniformly convex spaces, Transactions of the American Mathematical Society 40 (1936), no. 3, 396–396. CrossrefGoogle Scholar

  • [8] R. Espínola and A. Fernández-León, CAT(k)-spaces, weak convergence and fixed points, Journal ofMathematical Analysis and Applications 353 (2009), no. 1, 410–427. Google Scholar

  • [9] T. Foertsch, Ball versus distance convexity of metric spaces, Contributions to Algebra and Geometry (2004). Google Scholar

  • [10] R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math (1980). Google Scholar

  • [11] M. Kell, On Interpolation and Curvature via Wasserstein Geodesics, arxiv:1311.5407 (2013). Google Scholar

  • [12] W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Analysis: Theory, Methods&Applications 68 (2008), no. 12, 3689–3696. CrossrefGoogle Scholar

  • [13] K. Kuwae, Jensen’s inequality on convex spaces, Calculus of Variations and Partial Differential Equations 49 (2013), no. 3-4, 1359–1378. Google Scholar

  • [14] N. Monod, Superrigidity for irreducible lattices and geometric splitting, Journal of the American Mathematical Society 19 (2006), no. 4, 781–814. CrossrefGoogle Scholar

  • [15] A. Noar and L. Silberman, Poincaré inequalities, embeddings, and wild groups, Compositio Mathematica 147 (2011), no. 05, 1546–1572 (English). Google Scholar

  • [16] S. Ohta, Convexities of metric spaces, Geometriae Dedicata 125 (2007), no. 1, 225–250. Web of ScienceGoogle Scholar

  • [17] K.-Th. Sturm, Generalized Orlicz spaces and Wasserstein distances for convex-concave scale functions, Bulletin des Sciences Mathématiques 135 (2011), no. 6-7, 795–802. Google Scholar

  • [18] C. Villani, Optimal transport: old and new, Springer Verlag, 2009. Google Scholar

  • [19] T. Yokota, Convex functions and barycenter on CAT(1)-spaces of small radii, Preprint available at http://www.kurims.kyotou. ac.jp/˜takumiy/ (2013). Google Scholar

About the article

Received: 2014-07-07

Accepted: 2014-11-14

Published Online: 2014-12-10

Citation Information: Analysis and Geometry in Metric Spaces, Volume 2, Issue 1, ISSN (Online) 2299-3274, DOI: https://doi.org/10.2478/agms-2014-0015.

Export Citation

© 2014 Martin Kell. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Manuel De la Sen
Journal of Mathematics, 2017, Volume 2017, Page 1

Comments (0)

Please log in or register to comment.
Log in