Uniformly Convex Metric Spaces

Martin Kell

Analysis and Geometry in Metric Spaces (2014)

  • Volume: 2, Issue: 1, page 359-380, electronic only
  • ISSN: 2299-3274

Abstract

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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.

How to cite

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Martin Kell. "Uniformly Convex Metric Spaces." Analysis and Geometry in Metric Spaces 2.1 (2014): 359-380, electronic only. <http://eudml.org/doc/268793>.

@article{MartinKell2014,
abstract = {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.},
author = {Martin Kell},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {convex metric spaces; weak topologies; generalized barycenters; Banach-Saks property; ; convex metric space; Busemann curvature; convex subset; reflexivity; weak topology; barycenter; generalized convexity},
language = {eng},
number = {1},
pages = {359-380, electronic only},
title = {Uniformly Convex Metric Spaces},
url = {http://eudml.org/doc/268793},
volume = {2},
year = {2014},
}

TY - JOUR
AU - Martin Kell
TI - Uniformly Convex Metric Spaces
JO - Analysis and Geometry in Metric Spaces
PY - 2014
VL - 2
IS - 1
SP - 359
EP - 380, electronic only
AB - 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.
LA - eng
KW - convex metric spaces; weak topologies; generalized barycenters; Banach-Saks property; ; convex metric space; Busemann curvature; convex subset; reflexivity; weak topology; barycenter; generalized convexity
UR - http://eudml.org/doc/268793
ER -

References

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  11. [11] M. Kell, On Interpolation and Curvature via Wasserstein Geodesics, arxiv:1311.5407 (2013). 
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  16. [16] S. Ohta, Convexities of metric spaces, Geometriae Dedicata 125 (2007), no. 1, 225–250. Zbl1140.52001
  17. [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. Zbl1232.46028
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