TY - JOUR

T1 - Best approximation in polyhedral Banach spaces

AU - Fonf, Vladimir P.

AU - Lindenstrauss, Joram

AU - Veselý, Libor

N1 - Funding Information:
The first author was supported in part by the ISF grant 209/09 and by the Istituto Nazionale di Alta Matematica of Italy . The third author was supported in part by the Ministero dell’Istruzione, dell’Università e della Ricerca of Italy and by the Center for Advanced Studies in Mathematics at the Ben-Gurion University of the Negev, Beer-Sheva, Israel .

PY - 2011/11/1

Y1 - 2011/11/1

N2 - 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.

AB - 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.

KW - Metric projection

KW - Polyhedral Banach space

KW - Proximinal subspace

UR - http://www.scopus.com/inward/record.url?scp=80052669200&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2011.06.011

DO - 10.1016/j.jat.2011.06.011

M3 - Article

AN - SCOPUS:80052669200

SN - 0021-9045

VL - 163

SP - 1748

EP - 1771

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

IS - 11

ER -