Vol. 16, No. 1, 1966

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ISSN: 0030-8730
Families of parallels associated with sets

Eugene Elliot Robkin and F. A. Valentine

Vol. 16 (1966), No. 1, 147–157
Abstract

There exist sets S in Euclidean space En which have an interesting association with a family 𝒫 of parallel lines. For instance S and 𝒫 may be so related that each point of S lies on a member of 𝒫 which intersects S in either a line segment or a point. There exist compact sets S E2 such that every finite collection of points in S is contained in some collection of parallel lines each of which intersects S in a single point, and yet no infinite family 𝒫 of parallel lines exists having the same property and covering S. This paper contains a theorem which enables one to determine the existence of a family of parallel lines each of which intersects S in a line segment or point and which as a family covers S.

Secondly we show that the points, the closed line segments, the closed convex triangular regions, and the closed convex sets bounded by parallelograms are the only compact convex sets B in E2 which have the following property. If A is a closed connected set disjoint from B and if every 3 or fewer points of A lie on parallel lines intersecting B, then A is covered by a family of parallel lines each of which intersects B.

Finally, we obtain a theorem of Krasnoselskii type. Intuitively, this may be stated as follows. Suppose S is a compact set in En and suppose there exists a plane H such that every n points of S can see H via S along parallel lines. Then all the points of S can see H via S along parallel lines.

Mathematical Subject Classification
Primary: 52.34
Milestones
Received: 6 July 1964
Published: 1 January 1966
Authors
Eugene Elliot Robkin
F. A. Valentine