Metric projection bound and the Lipschitz constant of the radial retraction

Abstract

Let P M denote the metric projection on a proximinal subspace M of a real normed linear space X. Let ∥ P M ∥ = Sup{∥ y∥: y ϵ P M ( x), ∥ x∥ ⩽ 1}. It is shown that the Lipschitz constant for the radial retraction of the unit ball of X is equal to the metric projection bound, which is defined to be MPB( X) = Sup{∥ P M ∥: M proximinal subspace of X}. A formula for MPB( l 2 p ), 1 < p < ∞, is derived in the end.