Descriptive theory of nearest points in Banach spaces

Abstract

Let $X$ be a separable Banach space, $Y$ a closed, nonreflexive, linear subspace, and $P$ the set of points admitting a nearest approximation in $Y$. Then $P$ is an analytic set, and has three obvious algebraic properties. By adjusting the norm of $X$, any analytic set of this kind can be realized as the set of elements proximal to $Y$.