Porous sets for mutually nearest points in Banach spaces

Chong Li
Józef Myjak

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 \(\sigma\)-porous subset of \(\mathfrak{D}\).

Keywords: minimization problem, well-posedness, \(H_{\rho}\)-topology, \(\sigma\)-porous set.

Mathematics Subject Classification: 41A65, 54E52, 46B20.

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