CONVEX-SETS IN BANACH-SPACES AND A PROBLEM OF ROLEWICZ

Let B-X be the set of ail closed, convex and bounded subsets of a Bana ch space X equipped with the Hausdorff metric. In the first part of th is work we study the density character of B-X and investigate its conn ections with the geometry of the space, in particular with a property shared by the spaces of Shelah and Kunen. In the second part we are co ncerned with the problem of Rolewicz, namely the existence of support sets, for the case of spaces C(K).