Antiproximinal Norms in Banach Spaces

Abstract

We prove that every Banach space containing a complemented copy of c 0 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. © 2002 Elsevier Science (USA)

Relationships Between Farthest Point Problem and Best Approximation Problem

“…Also, some sufficient conditions for proximinality can be obtained using the closedness criterium established by Ka-Sing Lau in [20]. The antiproximinal norms was studied by Borwein, Jimenez-Sevilla and Moreno [8]. They obtained many existence results in some special spaces.…”

A partial positive solution to a conjecture of Ricceri

“…Concerning the existence of bounded convex anti-proximinal sets in Banach spaces see the papers [2], [4]. We remind the reader that a subset of a topological space is said to be rare exactly when its closure has empty interior.…”