Vol. 18, No. 2, 1966

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A convexity property

Raymond William Freese

Vol. 18 (1966), No. 2, 237–241

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).

Mathematical Subject Classification
Primary: 52.30
Secondary: 52.50
Received: 21 April 1965
Published: 1 August 1966
Raymond William Freese