On Generic Wellposedness of Restricted Chebyshev Center Problems in Banach Spaces

Let . (resp. K , .C, K C) denote the set of all nonempty bounded (resp. compact, bounded convex, compact convex) closed subsets of the Banach space X, endowed with the Hausdorff metric, and let G be a nonempty relatively weakly compact closed subset of X. Let .ostand for the set of all F . . such that the problem (F,G) is well.posed. We proved that, if X is strictly convex and Kadec, the set K C ..o is a dense G . .subset of K C \ G. Furthermore, if X is a uniformly convex Banach space, we will prove more, namely that the set .\.o (resp.K \.o, .C \.o, K C \.o) is ..porous in . (resp. K , .C, K C). Moreover, we prove that for most (in the sense of the Baire category) closed bounded subsets G of X, the set K \.o is dense and uncountable in K