Best approximation and intersections of balls in Banach spaces

Abstract

Let E be a Banach space, M a closed subspace of E with the 3-ball property. It is known that M is proximinal in E, and that its metric projection admits a continuous selection. This means that there is a continuous (generally non-linear) map π: E → M satisfying ‖x−π(x)‖ = d(x, M) for all x in E. Here it is shown that the same conclusion holds under a much weaker hypothesis on M, which we call the 1½-ball property. We also establish that if M has the 1½-ball property in E, then there is a continuous Hahn-Banach extension map from M* to E*.