What conditions determine when a collection of points A lies on a collection of parallel lines each member of which intersects a set B? In order to describe these conditions the following notations and definitions are used. Also for earlier results see Robkin and Valentine (2). Notations. We use the following abbreviations where E n is ^-dimensional Euclidean space and where S C E n , x £ E n , y G E n : cl 5 = closure of S, int 5 = interior of S, bd 5 = boundary of S, conv S = convex hull of

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... xy = closed line segment joining x and y when x ^ y, L (xy) = line determined by x and y when x ^ y, 0 = empty set, 0 = origin of S. The symbols \J, P\, and ~ are used for set union, set intersection, and set difference respectively. DEFINITION 1. A set of points A in E n has the m-point parallel line intersection property P(m) relative to a set B in E n if every collection of m or fewer points of A lies on a collection of parallel lines each member of which intersects B. The set A in E n is said to have the parallel line intersection property P(A) relative to the set B in E n if all the points of A lie on a collection of parallel lines each member of which intersects B. In this treatment we shall characterize those compact convex sets B in the plane £2 such that if A is a closed connected set in Ei which is disjoint from B and which has property P(m) relative to B, then A also has property P(A) relative to B (m is an integer). The concepts of "exposed point" and "antipodal points" play a crucial role in this development. DEFINITION 2. A point x in the boundary of a closed convex set B C E 2 is called an exposed point of S if there exists a line of support L to B such that LC\B = x.