Banach spaces antiproximinal in their biduals (Q1077653)

A subspace X of a Banach space Y is proximinal (antiproximinal) if every vector \(y\in Y\setminus X\) has a (has no) closest approximant in X. Most classical Banach spaces are proximinal in their second duals \(X^{**}\). It is shown that if X has a Schauder basis, then X can be renormed so that it is anti-proximinal in \(X^{**}\). Also, X can be renormed so that the compact operators \({\mathcal K}(X)\) are not proximinal in \({\mathcal B}({\mathcal H})\). (It cannot be antiproximinal.)