The Prox Map
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 156. 42X-443 [ 1991) The Prox Map GEKALU BEEK Deparrmenl of Marhemarics, Calij&niu Srare University, Los Angeles, Cahforniu 90032 AND DEVIDAS PAI * Department of Mathematics, Indian Invrituk qf’ Technolog.y, Powai, Bombay, 400076, India Submirred by R. P. Boas Recetved March 20, 198Y Let B be a weak*-compact subset of a dual normcd space X* and let A be a weak*-closed subset of X*. With d(B, A) denoting the distance between B and A in the usual sense, Prox(B, A) is the nonempty weak*-compact subset of BX A consisting of all (b, a) for which lib-al/ =d(B, A). In this article we study continuity properties of (B, A) + d(B, A) and (B, A) --* Prox(B, A), obtaining a generic theorem on points of single valuedness of the Prox map for convex sets. Best approximation and fixed point theorems for convex-valued multifunctions are obtained as applications. x 1991 Acddrmc Prraa. Inc. 1. INTRODUCTION Let X be a normed linear space. Given nonempty subsets A and B of X we write d(B,A) for inf{lIb-all :beB and UEA}. Points bEB and UEA are called proximal points of the pair (B, A) provided )(b - a/I = d(B, A). We denote the (possibly void) set of ordered pairs (h, a) of proximal points for (B, A) by Prox(B, A). Note that (6, a) E Prox(B, A) if an only if a-b is a point nearest the origin in A ~ B.
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