  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.
