Almost Chebyshev subspaces, lower semicontinuity, and Hahn-Banach extensions

Abstract

A subset $M$ of a Banach space is called almost Chebyshev iff the set of elements with more than one best approximation from $M$ is the first category. It is first shown that if the metric projection onto a proximinal almost Chebyshev subset $M$ is lower semicontinuous, then $M$ is Chebyshev. Next, let $M$ be a subspace of a separable Banach space. Then ${M^ \bot }$ is almost Chebyshev iff the set of elements in ${M^\ast }$ which fail to have a unique Hahn-Banach extension is the first category.