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.
@article{bwmeta1.element.doi-10_2478_agms-2014-0015,
author = {Martin Kell},
title = {Uniformly Convex Metric Spaces},
journal = {Analysis and Geometry in Metric Spaces},
volume = {2},
year = {2014},
zbl = {1311.53062},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_agms-2014-0015}
}
Martin Kell. Uniformly Convex Metric Spaces. Analysis and Geometry in Metric Spaces, Tome 2 (2014) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_agms-2014-0015/
[1] M. Bacák, Convex analysis and optimization in Hadamard spaces, Walter de Gruyter & Co., Berlin, 2014. | Zbl 1299.90001
[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. | Zbl 1322.53040
[3] J. B. Baillon, Nonexpansive mappings and hyperconvex spaces, Contemp. Math (1988). | Zbl 0653.54021
[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. | Zbl 0386.53048
[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). | Zbl 1158.49043
[7] J. A. Clarkson, Uniformly convex spaces, Transactions of the American Mathematical Society 40 (1936), no. 3, 396–396. | Zbl 0015.35604
[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. | Zbl 1182.47043
[9] T. Foertsch, Ball versus distance convexity of metric spaces, Contributions to Algebra and Geometry (2004). | Zbl 1070.53046
[10] R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math (1980). | Zbl 0505.46011
[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&Applications 68 (2008), no. 12, 3689–3696. | Zbl 1145.54041
[13] K. Kuwae, Jensen’s inequality on convex spaces, Calculus of Variations and Partial Differential Equations 49 (2013), no. 3-4, 1359–1378. | Zbl 1291.53047
[14] N. Monod, Superrigidity for irreducible lattices and geometric splitting, Journal of the American Mathematical Society 19 (2006), no. 4, 781–814. | Zbl 1105.22006
[15] A. Noar and L. Silberman, Poincaré inequalities, embeddings, and wild groups, Compositio Mathematica 147 (2011), no. 05, 1546–1572 (English). | Zbl 1267.20057
[16] S. Ohta, Convexities of metric spaces, Geometriae Dedicata 125 (2007), no. 1, 225–250. | Zbl 1140.52001
[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. | Zbl 1232.46028
[18] C. Villani, Optimal transport: old and new, Springer Verlag, 2009. | Zbl 1156.53003
[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).