Minimal sets of visibility

Abstract

Let S be a set in an n-dimensional Euclidean space, En. The following concept was used by Horn and Valentine [21 in their study of L sets, and it provides the basis of this investigation. DEFINITION 1. A set VCS is a set of visibility in S if, given any point p ES, there exists a point qE V such that the closed segment pqCS. NOTATION. Given a point xES, let V(x) denote a continuum' of visibility in S which contains x. The notation Vi(x) will also be used. DEFINITION 2. The set V(x) is a minimal of visibility in S relative to x if, for any other of visibility Vi(x), we have NlX) ?T V(x). 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 V(x) 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.