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

Abstract

Let ℬ (resp. ${\fancyscript K}$ , ℬ${\fancyscript C}$, ${\fancyscript K}$ ${\fancyscript C}$) 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^ostand 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 ${\fancyscript K}$ ${\fancyscript C}$ ∩ℬ^o is a dense G _δ –subset of ${\fancyscript K}$ ${\fancyscript C}$ \ G. Furthermore, if X is a uniformly convex Banach space, we will prove more, namely that the set ℬ\ℬ^o (resp.${\fancyscript K}$ \ℬ^o, ℬ${\fancyscript C}$ \ℬ^o, ${\fancyscript K}$ ${\fancyscript C}$ \ℬ^o) is σ–porous in ℬ (resp. ${\fancyscript K}$ , ℬ${\fancyscript C}$, ${\fancyscript K}$ ${\fancyscript C}$). Moreover, we prove that for most (in the sense of the Baire category) closed bounded subsets G of X, the set ${\fancyscript K}$ \ℬ^o is dense and uncountable in ${\fancyscript K}$.