A characterization of simply connected closed arcwise convex sets

Abstract

Let S be a set of points in the Euclidean plane E2. It is our purpose to establish a necessary and sufficient condition that a simply connected1 closed set S be arcwise convex. In order to do this precisely, the following notations and definitions are used. NOTATION. The line determined by two distinct points x and y in E2 is denoted by L(x, y). We designate the open line segment joining x and y by xy, and the corresponding closed segment by [xy]. The two closed half-planes having L(x, y) as a common boundary are designated by Ri(x, y) and R2(x, y). The boundary of a set K is represented by B(K), and H(K) denotes the convex hull of K. The complement of S is denoted by C(S). DEFINITION 1. A set SCE2 is said to be unilaterally connected if, for each pair of distinct points x and y in S, there exists a continuum2 Si CS which contains x and y, and which lies in one of the closed halfplanes determined by L(x, y). DEFINITION 2. A set SCE2 is said to be arcwise convex if each pair of points in S can be joined by a convex arc lying in S. (A convex arc is one which is contained in the boundary of its convex hull.) In a previous paper [1 ]3 the author studied the complements of both arcwise convex sets and unilaterally connected sets. The theorem below establishes another intimate connection between these two concepts. I am indebted to the referee for the following lemma which simplifies the proof of the theorem.