On Generic Well-posedness of Restricted Chebyshev Center Problems in Banach Spaces

Let B (resp. K , BC , 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 B stand for the set of all F ∈ B such that the problem (F,G) is well-posed. We proved that, if X is strictly convex and Kadec, the set K C ∩ B is a dense Gδ-subset of K C \ G. Furthermore, if X is a uniformly convex Banach space, we will prove more, namely that the set B \Bo (resp. K \Bo, BC \Bo, K C \Bo) is σ-porous in B (resp. K , BC , K C ). Moreover, we prove that for most (in the sense of the Baire category) closed bounded subsets G of X, the set K \ B is dense and uncountable in K .