@Article{Li2008,
author = "Li, Chong and Myjak, Józef",
title = "Porous sets for mutually nearest points in Banach spaces",
journal = "Opuscula Math.",
year = "2008",
volume = "28",
number = "1",
pages = "73--82",
doi = "http://dx.doi.org/10.7494/OpMath.2008.28.1.73",
url = "http://dx.doi.org/10.7494/OpMath.2008.28.1.73",
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",
issn = "1232-9274"
}