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Browsing by Subject "minimization problem"

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  • thumbnail.journal.alt
    Item type:Article, Access status: Open Access ,
    Porous sets for mutually nearest points in Banach spaces
    (2008) Li, Chong; Myjak, Józef
    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 σ-porous subset of $\mathfrak{D}$.