Three Point Arcwise Convexity

F. A. Valentine
1955 Proceedings of the American Mathematical Society  
Let 5 be a set in a two dimensional Euclidean space E2. Such a set 5 is said to be arcwise convex [5] if each pair of its points can be joined by a convex arc lying in S. A convex arc is, by definition, an arc which is contained in the boundary of a plane convex set. In two previous papers, [5; 6], the author studied certain properties of closed arcwise convex sets. It is the purpose of this paper to study an interesting class of sets which satisfy the three point arcwise convexity property,
more » ... vexity property, defined as follows: Definition 1. A set SQE2 is said to have the three point arcwise convexity property if each triple of points x£S, y £S, z(E.S is contained in a convex arc belonging to S. It should be observed that the above property implies there exists a convex arc in 5 having two of the three points x, y, z as end points and the third point in its interior. Definition 2. A convex curve, as distinguished from a convex arc, is a closed connected portion of the boundary of a plane convex set. A convex curve may have two, one, or no end points, and it may be bounded or unbounded. The following theorem characterizes the closed sets in E2 which have the three point arcwise convexity property. Theorem 1. Let S be a closed set in E2 which has at least three points. Then S has the three point arcwise convexity property if and only if it satisfies at least one of the following three conditions. 1. It is a closed convex set. 2. It is a convex curve. 3. It is a closed convex set except for one bounded convex hole, that is, it is obtained by deleting from a closed convex set a bounded open convex subset. In order to prove the necessity of Conditions 1-3, we shall establish four lemmas. In each of them it is assumed that S is a closed set which has at least three points, and which has the three point arcwise convexity property. Lemma 1. If x, y, z are three collinear points of S, with y between x and z, then at least one of the closed segments xy or yz belongs to S.
doi:10.2307/2032912 fatcat:7aoasyq3bvardaurcinayjrx2y