ABSTRACT. Closed convex bounded antiproximinal bodies are constructed in the infinite-dimensional spaces C(Q), Co(T), Loo(S, E,/~), and B(S), where Q is a topological space and T is a locally compact Hauedorff space. It is shown that there are no closed bounded antiproximim, l sets in Banach spaces with the RadonNikodym property.KEY WORDS: antiproximinal sets, convex bodies, Radon-Nikodym property.A subset A of a Banach space X is said to be antiproziminal if for any z E X \ A there is no point nearest to z in A.In the present paper, we construct antiproximinal closed convex bounded bodies in spaces of continuous functions and related spaces. The first example of a set of this kind was discovered by Edelstein and Thompson We use the following notation: X is a real Banach space; X* is the dual space of X; ~" = ~'(X) is the family of nonempty closed subsets of X; for x, y 9 X and M 9 ~', r > 0, we set xy = Hx -YH and
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