Geodesic
Parallel Lines Associated with Two Sets
Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 811-816

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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). 
Robkin, Eugene; Valentine, F. A. Parallel Lines Associated with Two Sets. Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 811-816. doi: 10.4153/CJM-1966-080-9
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[1] 1. Helly, E., Über Mengen konvexer Körper mit gemeinschaftlichen Punkten, Jber. Deutsch. Math. Verein., 32 (1923), 175–176. Google Scholar

[2] 2. Robkin, E. and Valentine, F. A., Families of parallel lines associated with sets, Pac. J. Math., 16 (1966), 147–157. Google Scholar

[3] 3. Valentine, F. A., Convex sets (New York, 1964). Google Scholar

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