Well-posedness and porosity in best approximation problems

Abstract

Given a nonempty closed subset $A$ of a Banach space $X$ and a point $x \in X$, we consider the problem of finding a nearest point to $x$ in $A$. We define an appropriate complete metric space $\mathcal M$ of all pairs $(A,x)$ and construct a subset $\Omega$ of $\mathcal M$ which is the countable intersection of open everywhere dense sets such that for each pair in $\Omega$ this problem is well-posed. As a matter of fact, we show that the complement of $\Omega$ is not only of the first category, but also sigma-porous.