Banach spaces not antiproximinal in their second dual (Q807899)

A subspace M of a Banach space X is said to be antiproximinal in X if the only vectors in X which have closest proximinants from M are elements of M. In [\textit{K. R. Davidson}, J. Approximation Theory 47, 203-213 (1986; Zbl 0595.41034)], the author claims that if a Banach space X has the projection approximation property, then X has an equivalent norm \(| \cdot |\) under which it is antiproximinal in \(X^{**}\). The authors point out a gap in Davidson's argument and show that, in fact, \((\ell_ 1,| \cdot |)\) is not antiproximinal in its second dual.