A Motzkin-type theorem for closed nonconvex sets

Introduction. Bouligand [l] recognized the importance of the nearest-points mapping for a closed set X and the set Sx of points with more than one nearest point in X for the study of geometry. Later Motzkin [3], [4] used them in the proof of his theorem characterizing closed convex sets. We use them to show, essentially, that Sx characterizes the complement of X in its convex hull. Our result includes the Motzkin theorem as a special case and yields a theorem of Valentine [5] as a corollary. The original motivation and background for our work can be found in [2].