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In this paper the theory of uniformly convex metric spaces is developed. These spaces exhibit ageneralized convexity of the metric from a fixed point. Using a (nearly) uniform convexity property a simpleproof of reflexivity is presented and a weak topology of such spaces is analyzed. This topology, called coconvextopology, agrees with the usually weak topology in Banach spaces. An example of a CAT(0)-spacewith weak topology which is not Hausdorff is given.In the end existence and uniqueness of generalized barycenters is shown, an application to isometric groupactions is given and a Banach-Saks property is proved.

In this article, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel-Brascamp-Lieb inequality for general Riemannian and Finsler manifolds and led Lott-Villani and Sturm to define an abstract Ricci curvature condition. Following their ideas, a similar condition can be defined and for positively curved spaces one can prove a Poincaré inequality. Using Gigli's recently developed calculus on metric measure spaces, even a q-Laplacian comparison theorem holds on q-infinitesimal convex spaces. In the appendix, the theory of Orlicz-Wasserstein spaces is developed and necessary adjustments to prove the interpolation inequality along geodesics in those spaces are given. [ABSTRACT FROM AUTHOR]