Uniformly convex metric spaces

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 co-convex topology agrees with the usualy weak topology in Banach spaces. An example of a $CAT(0)$-spaces with weak topology which is not Hausdorff is given. This answers questions raised by Monod 2006, Kirk and Panyanak 2008 and Espínola and Fernández-León 2009. In the end existence and uniqueness of generalized barycenters is shown and a Banach-Saks property is proved.

23 pages

References in corpus (2)

• Poincaré inequalities, embeddings, and wild groups
• A notion of nonpositive curvature for general metric spaces

Cited by in corpus (2)

• Weak topology on CAT(0) spaces
• On a weak topology for Hadamard spaces and its applications