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.

Best approximation in polyhedral Banach spaces / V.P. Fonf, J. Lindenstrauss, L. Vesely. - In: JOURNAL OF APPROXIMATION THEORY. - ISSN 0021-9045. - 163:11(2011 Nov), pp. 1748-1771. [10.1016/j.jat.2011.06.011]

Best approximation in polyhedral Banach spaces

L. Vesely
Ultimo
2011

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.
Polyhedral Banach space ; Metric projection ; Proximinal subspace
Settore MAT/05 - Analisi Matematica
nov-2011
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/161232
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