Opuscula Math. 28, no. 1 (2008), 73-82
Opuscula Mathematica
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 \(\sigma\)-porous subset of \(\mathfrak{D}\).
Keywords: minimization problem, well-posedness, \(H_{\rho}\)-topology, \(\sigma\)-porous set.
Mathematics Subject Classification: 41A65, 54E52, 46B20.
- 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
- Received: 2007-04-02.
- Revised: 2007-06-09.
- Accepted: 2007-06-12.
