Vol. 21, No. 3, 1967

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On the union of two starshaped sets

D. G. Larman

Vol. 21 (1967), No. 3, 521–524

Let S be a compact subset of a topological linear space. We shall say that S has the property φ if there exists a line segment R such that each triple of points x,y and z in S determines at least one point p of R (depending on x,y and z) such that at least two of the segments xp,yp and zp are in S. It is clear that if S is the union of two starshaped sets then S has the property φ, and the problem has been raised by F. A. Valentine [1] as to whether the property φ ensures that S is the union of two starshaped sets. We shall show that this is not so, in general, but we begin by giving a further constraint which ensures the result.

Mathematical Subject Classification
Primary: 52.30
Received: 22 December 1965
Revised: 2 May 1966
Published: 1 June 1967
D. G. Larman