Antiproximinal sets in spaces of continuous functions

Abstract

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

On nearest and furthest points

“…Concerning convex antiproximinal sets, see [1] and the literature cited therein. Concerning sets with convex complement, see, for example, [2], where it is shown that there are no antiproximinal sets with smooth complement in reflexive Banach spaces.…”

Antiproximinal Sets in Banach Spaces of Continuous Vector-Valued Functions

“…Since all these terms belong to S λ + , it follows The neighborhood W of t 0 u 0 contains an infinity of terms of the form t 2m u 2m and an infinity of terms of the form t 2n+1 u 2n+1 as well. An analysis of the signs of the expressions 20 − 21α and 31α − 30 (see [2]) shows that in all of the cases S λ − ∩ S λ + = which, by Corollary 2.4, implies ∈ λ . Indeed, if α ∈ 0 30/31 then λ t 2n+1 u 2n+1 < 0 and λ t 2m , u 2m > 0 so that t 0 u 0 ∈ S λ − ∩ S λ + = S λ − ∩ S λ + If α ∈ 30/31 1 then λ t 0 u 0 < 0 and λ t 2m u 2m > 0 so that t 0 u 0 ∈ S λ − ∩ S λ + = S λ − ∩ S λ + Theorem 3.3 is completely proved.…”

“…The existence of antiproximinal bounded closed convex bodies in C T for more general compact spaces T , including 0 1 n , the Cantor perfect set, and the Hilbert cube, was proved by Fonf [14]. Recently, Balaganskii [2] proved that the space C T is of N 2 -type for an arbitrary infinite compact Hausdorff space T . In [8,10,11] it was proved that the vector-valued sequence spaces c 0 E c E C 1 α E , for E a Banach space and α a countable ordinal, are of N 2 -type too.…”

“…The aim of the present paper is to show that the device used by Balaganskii [2] to prove the existence of an antiproximinal bounded closed convex body in C T can be adapted to do the same thing for the vectorvalued version C T E of C T . Because the main tool used in the study of antiproximinality of a closed convex subset Z of a Banach space X is a relation between the support functionals of the set Z and those of the unit ball of X (Proposition 3.1 below), we shall study first the behavior of the support functionals of the unit ball of the space C T E .…”

“…Concerning the space L 1 T µ it was shown that it is not of N 2 -type if the measure space T µ contains at least one atom [5], and that the space L 1 −π π contains a radially bounded antiproximinal convex body [2].…”