Antiproximinal norms in Banach spaces



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Borwein, Jonathan M. and Jiménez Sevilla, María del Mar and Moreno, José Pedro (2002) Antiproximinal norms in Banach spaces. Journal of Approximation Theory, 114 (1). pp. 57-69. ISSN 0021-9045

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We prove that every Banach space containing a complemented copy of c0 has an antiproximinal body for a suitable norm. If, in addition, the space is separable, there is a pair of antiproximinal norms. In particular, in a separable polyhedral space X, the set of all (equivalent) norms on X having an isomorphic antiproximinal norm is dense. In contrast, it is shown that there are no antiproximinal norms in Banach spaces with the Convex Point of Continuity Property (CPCP). Other questions related to the existence of antiproximinal bodies are also discussed.

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This work was begun while the second and third authors were visiting the CECM at the Simon Fraser University. The second author is indebted to Gilles Godefroy for his support, valuable suggestions, and many stimulating conversations.

Uncontrolled Keywords:Convex-functions; Intersection-properties; Continuity property; Differentiability; Sets; Point
Subjects:Sciences > Mathematics > Numerical analysis
ID Code:16414
Deposited On:18 Sep 2012 08:39
Last Modified:27 Jun 2018 09:55

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