Uniformly convex metric spaces

TL;DR

This paper develops the theory of uniformly convex metric spaces, establishing properties like reflexivity, weak topology, and barycenter existence, with implications for non-Hausdorff spaces and general convexity concepts.

Contribution

It introduces a generalized convexity framework for metric spaces, proves reflexivity and weak topology properties, and provides examples and results on barycenters and Banach-Saks property.

Findings

Reflexivity of uniformly convex metric spaces proved.

Weak topology called co-convex topology analyzed.

Existence and uniqueness of generalized barycenters established.

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 Esp\'inola and Fern\'andez-Le\'on 2009. In the end existence and uniqueness of generalized barycenters is shown and a Banach-Saks property is proved.