TY  - JOUR
ID  - Li2008
LB  - Li2008
AU  - Li, Chong
AU  - Myjak, Józef
TI  - Porous sets for mutually nearest points in Banach spaces
ST  - Porous sets for mutually nearest points in Banach spaces
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2008
VL  - 28
IS  - 1
SP  - 73
EP  - 82
DO  - http://dx.doi.org/10.7494/OpMath.2008.28.1.73
UR  - http://dx.doi.org/10.7494/OpMath.2008.28.1.73
AB  - 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 \(\mathfrak{D}\).
KW  - minimization problem
KW  - well-posedness
KW  - \(H_{\rho}\)-topology
KW  - \(\sigma\)-porous set
SN  - 1232-9274
ER  -