Porous sets for mutually nearest points in Banach spaces

Abstract

Let $\mathfrak{B}(X)$ denote the family of all nonempty closed bounded subsets of a real Banach space $X$, endowed with the Hausdorff metric. For $E, F \in \mathfrak{B}(X)$ we set $\lambda_{EF} = \inf {|z - x| : x \in E, z \in F }$. Let $\mathfrak{D}$ denote the closure (under the maximum distance) of the set of all $(E, F) \in \mathfrak{B}(X) \times \mathfrak{B}(X)$ such that $\lambda_{EF} \gt 0$. It is proved that the set of all $(E, F) \in \mathfrak{D}$ for which the minimization problem $\min_{x \in E, z\in F}|x - z|$ fails to be well posed in a σ-porous subset of $\mathfrak{D}$.