A convexity property
Raymond W. Freese · Pacific Journal of Mathematics · 1966
There exist a variety of conditions yielding convexity of a set, dependent upon the nature of the underlying space.It is the purpose here to define a particular restriction involving ^-tuples (the ^-isosceles property) on subsets of a straight line space and study the effect of this restriction in establishing convexity.By a straight line space is meant a finitely compact, convex, externally convex metric space in which the linearity of two triples of a quadruple implies the linearity of the remaining two.The principal theorem states that the ^-isosceles property is a sufficient condition for a closed and arcwise connected subset of a straight line space to be convex if and only if n is two or three.In such a space S we use two of the definitions stated by Marr and Stamey (4).DEFINITION 1.If p, q, r are distinct points of S such that at least two of the distances pq, pr, qr are equal, then the points p, q f r are said to form an isosceles triple in S.DEFINITION 2. A subset M of S is said to have the double-isosceles three-point property if two connecting segments of each of its isosceles triples belong to M.A proof of (2) together with (4) shows that if M is a closed •connected subset of S and possesses the double isosceles property, then M is convex.DEFINITION 3. A subset M of S is said to have the w-isosceles property (n ^ 2) provided for every (n + l)-tuple p u p 2 , , p n+1 of distinct points of M such that PiP i+1 = p i+1 p i+ 2, i = 1, 2, , n -1, at least n of the connecting segments lie in M.