Support, Convexity Conditions and Convex Hypersurfaces in Infinite Dimension

Working in infinite dimensional linear spaces, we deal with support for closed sets without interior. We generalize the Convexity Theorem for closed sets without interior. Finally we study the infinite dimensional version of Jordan hypersurfaces. Our whole work never assumes smoothness and is based exclusively on non-differential Convex Analysis tools and, in particular, on theory of convex cones. A crucial mathematical tool for our results is obtained solving the decomposition problem for non-closed non pointed cones.