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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.
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