Antiproximinal sets in spaces of continuous functions on compacta (Q792584)

A subset A of a Banach space X is called antiproximinal, if to no point \(x\in X-A\) there exists the nearest point in A. Let a compact set K be expressed as a union of closed subsets \(K_ i\subset K\), \(i=1,2,..\). with the following property: There exist an increasing sequence of natural numbers \(n_ j\) and a collection of homeomorphic inclusion maps \(\phi_ j^{(i)}:K_ i\to K_{n_ j} (1\leq i<n_ j\), \(j=1,2,...)\) such that \[ \phi_ j^{(i)}(K_ i)\cap K_ p=\emptyset \quad for\quad p\neq n_ j \] and such that for every \(i=1,2,..\). and for every \(t\in K_ i\) the sequence \(\{\phi_ j^{(i)}(t)\}^{\infty}_{j=1}\) converges. It is proved that in the space C(K) of functions continuous on a compact set K satisfying the condition above there exists a sequence of antiproximinal sets which are convex and symmetric with respect to the origin. The same property is proved for Banach spaces X whose duals are isometric to the space \(\ell_ 1\).