Intersections of maximal starshaped sets
William R. Hare, John W. Kenelly · Proceedings of the American Mathematical Society · 1968
0. Introduction. In Valentine [1, p. 183] the problem of characterizing starshaped sets in terms of maximal convex sets was posed. One published solution says that the convex kernel of a set is the intersection of all the maximal convex subsets of the set [2, p. 280]. In this paper we investigate the analogous problem of describing the intersection of all maximal starshaped subsets of a set. A maximal starshaped subset X of a set Y is a starshaped subset of Y which is not properly contained in any other starshaped subset of Y. Since the property of being starshaped is not an intersectional property, it seems unlikely that the intersection of maximal starshaped subsets of a given set would be starshaped. Indeed, the following example shows the situation to be even mnore complex than merely absence of the intersectional property. Let Tn={(x, y)ln-1 y<n, n-x_y}, and Sn=U Un Ti; then Sn is starshaped with convex kernel, ck(Sn), equal to K. = { (x, y) I O <?y?_i1, n-x <y }. If S = Un 1 Sn, then ck(S) CU= I ck(Sn) =0. Thus S is not starshaped even though it is the union of an ascending chain of starshaped sets. Furthermore, S has no maximal starshaped subsets. If MCS were a maximal starshaped subset, then there would be at least one point (x, y) Eck(M). In fact M would be precisely the set of points that (x, y) sees via S. However, the point (x+1, y) sees every point which (x, y) does, and more. Thus M is not maximal. In contrast with the preceding example, it is shown in ?1 that compact subsets of Euclidean space, En, have maximal starshaped subsets. In ?2, it is shown that the intersection of the maximal starshaped subsets in a suitably restricted setting is starshaped.