Minimal Sets of Visibility

Let 5 be a set in an «-dimensional Euclidean space, E„. The following concept was used by Horn and Valentine [2] in their study of L sets, and it provides the basis of this investigation. Definition 1. A set VES is a set of visibility in S if, given any point p ES, there exists a point qEV such that the closed segment pqQS. Notation. Given a point xES, let Vix) denote a continuum1 of visibility in 5 which contains x. The notation V,ix) will also be used. Definition 2. The set Vix) is a minimal continuum of visibility in S relative to x if, for any other continuum of visibility 7i(x), we have ViixXtVix). A corresponding definition holds if we replace the word "continuum" by the words "compact convex set." It is our purpose to investigate sets for which Vix) is unique for each xES. The most interesting result is contained in Theorem 2. The corresponding theory in which maximal convex sets are considered has been developed by Strauss and Valentine [3]. The two theories are decidedly different, and this difference is explained at the end of this article.