Properties of typical bounded closed convex sets in Hilbert space

Abstract

For a nonempty separable convex subset $X$ of a Hilbert space $ℍ\left(\text{Ω}\right)$, it is typical (in the sense of Baire category) that a bounded closed convex set $C\subset ℍ\left(\text{Ω}\right)$ defines an $m$-valued metric antiprojection (farthest point mapping) at the points of a dense subset of $X$, whenever $m$ is a positive integer such that $m\le dimX+1$.