Best approximation by closed sets in Banach spaces

Abstract

The set K is called proximinal (Chebyshev) if every point x E X has a (unique) best approximation from K. It is easy to see that every closed convex set K in a reflexive space X is proximinal. In addition, if the norm is strictly convex, then K is Chebyshev. However, if X is not assumed reflexive or K is not assumed convex, then the above result is false in general. In [7], SteCkin introduced the concept of almost Chebyshev. A set K is called almost Chebyshev if the set of x in X such that K fails to have unique best approximation to x is a first category subset of X. He proved that if X is a uniformly convex Banach space, then every closed subset is almost Chebyshev. By using this concept, Garkavi [4] showed that for any reflexive subspace F in a separable Banach space, there exists a (in fact, many) subspace G which is B-isomorphic to F and is almost Chebyshev. The author [6] showed that if X is a separable Banach space which is locally uniformly convex or possesses the Radon-Nikodym property, then “almost all” closed subspaces are almost Chebyshev. In [3], Edelstein proved that if X has the Radon-Nikodym property, then for any bounded closed convex subset K, the set of x in X which admit best approximations from K is a weakly dense subset in X. In this paper, we generalize SteEkin’s result to a wider class of Banach spaces. A Banach space is called a U-space if for any E > 0, there exists 6 > 0 such that for any x, y E X with /I x ij = 1~ y il = 1 and 11(x + y)/2 (1 > 1 6, 11(x* + y*)/2 11 > 1 E, where x* and y* are norm 1 support functionals of the closed unit ball of X at x, y, respectively. We show that this class of spaces is self-dual, it contains all uniformly convex spaces,