Best approximation in polyhedral Banach spaces

Vladimir P. Fonf, Joram Lindenstrauss, Libor Veselý

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


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
Issue number11
StatePublished - 1 Nov 2011


  • Metric projection
  • Polyhedral Banach space
  • Proximinal subspace

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • General Mathematics
  • Applied Mathematics


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