Journal of Convex Analysis 19 (2012), No. 4, 1153--1166
Copyright Heldermann Verlag 2012
Best Approximation Problems in Compactly Uniformly Rotund Spaces
Julian P. Revalski
Inst. of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street - Block 8, 1113 Sofia, Bulgaria
revalski@math.bas.bg
Nikolay V. Zhivkov
Inst. of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street - Block 8, 1113 Sofia, Bulgaria
niz@abv.bg
We investigate under what geometric conditions the best approximation problem to a nonempty closed subset of a real Banach space is generalized well-posed, or, more generally, the problem either has no solution or is generalized well-posed, for the majority of the points in the space. "Majority" is understood as a set whose complement in the space is σ-porous or σ-cone supported. Analogously to the case when uniqueness of the best approximation is considered, it turns out that certain local uniform, or uniform, properties of the norm of the underlying space have to be required.
Keywords: Best approximation, metric projection, well-posedness, approximative compactness, Baire category, porous sets, cone supported sets, compact uniform rotundity.
MSC: 41A65, 46B20
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