On the Radial Projection in Normed Spaces

Our concern is with the Lipschitz constant of T; i.e. with the constant K such that || J # — 7 j | | ^i£||#—;y|| for all x, y in X. In particular, we wish to determine under what conditions on the space X the mapping T will be nonexpansive, i.e. K = l. T is a special case of a proximity mapping defined by a convex set in a normed vector space, i.e. a mapping which assigns to each point of X, the nearest point of the convex set C. There has been a good deal of interest in recent years in proximity maps, nonexpansive mappings, and their interrelations (Moreau, Browder, Petryshyn, Kirk, De Prima, Lions and Stampacchia, and others). I t is easy to see that if X is an inner product space (and in particular, a Hilbert space), then T and every proximity map is nonexpansive. More precisely, Kirk and Smiley [ l] proved that X is an inner product space if and only if for all nonzero x and y in X