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Small sets in best approximation theory

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Abstract

The best approximation problem to a nonempty closed set in a locally uniformly convex Banach space is considered. The main result states that the set of points which have best approximation but the approximation problem is not well-posed is very small in a sense that it is σ-cone supported in the underlying space. This gives an improvement of an original result of Stečkin about the set of points with more than one best approximation which involves Baire categories. Examples on the necessity of some of the imposed conditions are provided.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, block 8, 1113, Sofia, Bulgaria

    J. P. Revalski & N. V. Zhivkov

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  1. J. P. Revalski
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  2. N. V. Zhivkov
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Correspondence to J. P. Revalski.

Additional information

The authors have been partially supported by the Bulgarian National Fund for Scientific Research under contract No: DO02-360/2008.

Part of these results were obtained while the first named author was professeur associé in the group LAMIA in the Department of Mathematics and Informatics of Université des Antilles et de la Guyane, Guadeloupe, France.

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Revalski, J.P., Zhivkov, N.V. Small sets in best approximation theory. J Glob Optim 50, 77–91 (2011). https://doi.org/10.1007/s10898-010-9621-x

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