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 n-tuples (the
n-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 n-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).
