Best approximations and porous sets

Let D be a nonempty compact subset of a Banach space X and denote by S(X) the family of all nonempty bounded closed convex subsets of X. We endow S(X) with the Hausdorff metric and show that there exists a set F ⊂ S(X) such that its complement S(X) \ F is σ-porous and such that for each A ∈ F and each x̃ ∈ D, the set of solutions of the best approximation problem ‖x̃− z‖ → min, z ∈ A, is nonempty and compact, and each minimizing sequence has a convergent subsequence.