Modulus of Strong Proximinality and Continuity of Metric Projection

In this paper we initiate a quantitative study of strong proximinality. We define a quantity I mu(x, t) which we call as modulus of strong proximinality and show that the metric projection onto a strongly proximinal subspace Y of a Banach space X is continuous at x if and only if epsilon(x, t) is continuous at x whenever t > 0. The best possible estimate of epsilon(x, t) characterizes spaces with 1 1/2 ball property. Estimates of epsilon(x, t) are obtained for subspaces of uniformly convex spaces and of strongly proximinal subspaces of finite codimension in C(K).