Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Best approximation
  • metric projection
  • well-posedness
  • approximative compactness
  • Baire category
  • porous sets
  • cone supported sets
  • compact uniform rotundity

Mathematics Subject Classification

41A65, 46B20

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Suggested citation

J. P. Revalski, N. V. Zhivkov. “Best Approximation Problems in Compactly Uniformly Rotund Spaces.” Journal of Convex Analysis 19 (2012), No. 4, 1153–1166. https://doi.org/10.68381/jca19061

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.