Best approximation in polyhedral Banach spaces

Vladimir P. Fonf, Joram Lindenstrauss, Libor Veselý

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9 Scopus citations

Abstract

In the present paper, we study conditions under which the metric projection of a polyhedral Banach space X onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if X satisfies (*) (a geometric property stronger than polyhedrality) and Y⊂X is any proximinal subspace, then the metric projection PY is Hausdorff continuous and Y is strongly proximinal (i.e., if {yn}⊂Y, x∈X and ∥yn-x∥→dist(x,Y), then dist(yn,PY(x))→0).One of the main results of a different nature is the following: if X satisfies (*) and Y⊂X is a closed subspace of finite codimension, then the following conditions are equivalent: (a) Y is strongly proximinal; (b) Y is proximinal; (c) each element of Y⊥ attains its norm. Moreover, in this case the quotient X/Y is polyhedral.The final part of the paper contains examples illustrating the importance of some hypotheses in our main results.

Original languageEnglish
Pages (from-to)1748-1771
Number of pages24
JournalJournal of Approximation Theory
Volume163
Issue number11
DOIs
StatePublished - 1 Nov 2011

Keywords

  • Metric projection
  • Polyhedral Banach space
  • Proximinal subspace

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