Proximity maps for convex sets

The method of successive approximation is applied to the problem of obtaining points of minimum distance on two convex sets. Specifically, given a closed convex set K in Hilbert space, let P be the map which associates with each point x the point Px of K closest to x. That P is well-defined is proved in [l, p. 6]. P will be called the proximity map for K. If there are two such sets, Ki and Kt, let Q denote the composition P1P2 of their proximity maps. It is shown that every fixed point of Q is a point of Ki closest to K2, and that the fixed points of Q may be obtained by iteration of Q when one of the sets is compact or when both are polytopes in E„. An application to the solution of linear inequalities is cited. Our thanks are due the referee for having suggested substantial simplifications.