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CHEBYSHEV SETS

Part of: Approximations and expansions Miscellaneous applications of operator theory Functional analysis Normed linear spaces and Banach spaces; Banach lattices Nonlinear operators and their properties

Published online by Cambridge University Press:  11 November 2014

JAMES FLETCHER  and
WARREN B. MOORS
JAMES FLETCHER
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong 2522, Australia email [email protected]
WARREN B. MOORS*
Affiliation:
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand email [email protected]
*
✉[email protected]
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Abstract

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A Chebyshev set is a subset of a normed linear space that admits unique best approximations. In the first part of this paper we present some basic results concerning Chebyshev sets. In particular, we investigate properties of the metric projection map, sufficient conditions for a subset of a normed linear space to be a Chebyshev set, and sufficient conditions for a Chebyshev set to be convex. In the second half of the paper we present a construction of a nonconvex Chebyshev subset of an inner product space.

Keywords

convex setsChebyshev setsBanach spacesbest approximationsmetric projectionsuniquely remotal sets

MSC classification

Primary: 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Secondary: 41A50: Best approximation, Chebyshev systems 46B20: Geometry and structure of normed linear spaces 47N10: Applications in optimization, convex analysis, mathematical programming, economics 47H05: Monotone operators and generalizations 46-02: Research exposition (monographs, survey articles)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 98 , Issue 2 , April 2015 , pp. 161 - 231
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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