Metric projections after renorming

Abstract

A multi-valued mapping of a reflexive real Banach space into its subspace is a metric projection for a suitable equivalent norm iff it has non-empty closed convex values, is norm-to-weak upper semi-continuous, and is semi-linear. As an application of this characterization we prove that, given an infinite-dimensional subspace of codimension at least two in a reflexive space, there exists an equivalent norm such that the subspace is Chebyshev but the metric projection is not continuous.