Antiproximal sets (Q1822291)

A set C in a Banach space X is said to be antiproximal if no point in \(X\setminus C\) has a nearest point in S. If C is a closed bounded symmetric convex body in X, then, it is readily seen, C is antiproximal if and only if the closed unit ball of X is antiproximal in the norm induced by C. Denote by U (respectively D) the unit ball of \(c_ 0\) equipped with the usual norm (respectively Day's norm). \textit{S. Cobzaş} [Math. Rev. Anal. Numér. Téor. Approximation, Math. 2, 137-141 (1973)] has proved that D is antiproximal in \(c_ 0\) with respect to the usual norm. In the present paper the author constructs an isomorphism \(A: c_ 0\to c_ 0\), \(A\neq id.\), such that A(U) is antiproximal in \(c_ 0\) with respect to Day's norm.