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.
