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Approximation properties of sets with bounded complements

Published online by Cambridge University Press:  14 November 2011

C. Franchetti  and
P. L. Papini
C. Franchetti
Affiliation:
University of Genova, Italy
P. L. Papini
Affiliation:
University of Bologna, Italy
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Synopsis

Given a Banach space X, we investigate the behaviour of the metric projection PF onto a subset F with a bounded complement.

We highlight the special role of points at which d(x, F) attains a maximum. In particular, we consider the case of X as a Hilbert space: this case is related to the famous problem of the convexity of Chebyshev sets.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 89 , Issue 1-2 , 1981 , pp. 75 - 86
Copyright
Copyright © Royal Society of Edinburgh 1981

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References

1Asplund, E.. Čebyšev sets in Hilbert spaces. Trans. Amer. Math. Soc. 144 (1969), 235–240.Google Scholar
2Berens, H. and Westphal, U.. Kodissipative metrische projektionen in normierten linearen räumen. In Linear spaces and approximation 119–130, ed. Butzer, P. L. and Sz-Nagy, B., ISNM Vol. 40 (Basel: Birkhäuser, 1978).Google Scholar
3Diestel, J.. Geometry of Banach spaces – Selected topics. Lecture Notes in Mathematics 485 (Berlin: Springer, 1975).Google Scholar
4Franchetti, C. and Singer, I.. Best approximation by elements of caverns in normed linear spaces. Boll. Un. Mat. Ital. 17–B (1980), 33–43.Google Scholar
5Godini, G.. On suns. Rev. Roumaine Math. Pures Appl. 23 (1978), 893–903.Google Scholar
6Langenshop, C. E.. Differentiability of the distance to a set in Hilbert space. J. Math. Anal. Appl. 44 (1973), 620–624.Google Scholar
7Papini, P. L.. Sheltered points in normed spaces. Ann. Mat. Pura Appl. 117 (1978), 233–242.Google Scholar
8Yaglom, I. M. and Boltyanskiǐ, V. G.. Convex figures (English translation from the Russian original) (New York: Holt, Rinehart and Winston, 1961).Google Scholar