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Tytuł artykułu

Uniformly Convex Metric Spaces

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Rocznik
Tom
2
Numer
1
Opis fizyczny
Daty
otrzymano
2014-07-07
zaakceptowano
2014-11-14
online
2014-12-10
Twórcy
autor
  • Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany, mkell@mis.mpg.de
Bibliografia
  • [1] M. Bacák, Convex analysis and optimization in Hadamard spaces, Walter de Gruyter & Co., Berlin, 2014.
  • [2] M. Bacák, B. Hua, J. Jost, M. Kell, and A. Schikorra, A notion of nonpositive curvature for general metric spaces, DifferentialGeometry and its Applications 38 (2015), 22–32.
  • [3] J. B. Baillon, Nonexpansive mappings and hyperconvex spaces, Contemp. Math (1988).
  • [4] M. R. Bridson and A. Häfliger, Metric Spaces of Non-Positive Curvature, Springer, 1999.
  • [5] H. Busemann and B. B. Phadke,Minkowskian geometry, convexity conditions and the parallel axiom, Journal of Geometry 12(1979), no. 1, 17–33.
  • [6] Th. Champion, L. De Pascale, and P. Juutinen, The1-Wasserstein Distance: Local Solutions and Existence of Optimal TransportMaps, SIAM Journal on Mathematical Analysis 40 (2008), no. 1, 1–20 (en).
  • [7] J. A. Clarkson, Uniformly convex spaces, Transactions of the American Mathematical Society 40 (1936), no. 3, 396–396.
  • [8] R. Espínola and A. Fernández-León, CAT(k)-spaces, weak convergence and fixed points, Journal ofMathematical Analysis andApplications 353 (2009), no. 1, 410–427.
  • [9] T. Foertsch, Ball versus distance convexity of metric spaces, Contributions to Algebra and Geometry (2004).
  • [10] R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math (1980).
  • [11] M. Kell, On Interpolation and Curvature via Wasserstein Geodesics, arxiv:1311.5407 (2013).
  • [12] W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Analysis: Theory, Methods&Applications68 (2008), no. 12, 3689–3696.
  • [13] K. Kuwae, Jensen’s inequality on convex spaces, Calculus of Variations and Partial Differential Equations 49 (2013), no. 3-4,1359–1378.
  • [14] N. Monod, Superrigidity for irreducible lattices and geometric splitting, Journal of the American Mathematical Society 19(2006), no. 4, 781–814.
  • [15] A. Noar and L. Silberman, Poincaré inequalities, embeddings, and wild groups, Compositio Mathematica 147 (2011), no. 05,1546–1572 (English).
  • [16] S. Ohta, Convexities of metric spaces, Geometriae Dedicata 125 (2007), no. 1, 225–250.
  • [17] K.-Th. Sturm, Generalized Orlicz spaces and Wasserstein distances for convex-concave scale functions, Bulletin des SciencesMathématiques 135 (2011), no. 6-7, 795–802.
  • [18] C. Villani, Optimal transport: old and new, Springer Verlag, 2009.
  • [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).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_agms-2014-0015
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