Opuscula Math. 28, no. 1 (2008), 73-82

Opuscula Mathematica

# 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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• Chong Li
• Zhejiang University, Department of Mathematics, Hangzhou 310027, P. R. China
• Józef Myjak
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Cracow
• Revised: 2007-06-09.
• Accepted: 2007-06-12.