Sylvester's problem and Motzkin's theorem for countable and compact sets

Author:
Peter B. Borwein

Journal:
Proc. Amer. Math. Soc. **90** (1984), 580-584

MSC:
Primary 52A37

DOI:
https://doi.org/10.1090/S0002-9939-1984-0733410-4

MathSciNet review:
733410

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Abstract | References | Similar Articles | Additional Information

Abstract: The following three variations of Sylvester's Problem are established. Let and be compact, countable and disjoint sets of points.

(1) If spans (the Euclidean plane) then there must exist a line through two points of that intersects in only finitely many points.

(2) If spans (Euclidean three-space) then there must exist a line through exactly two points of .

(3) If spans then there must exist a line through at least two points of one of the sets that does not intersect the other set.

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DOI:
https://doi.org/10.1090/S0002-9939-1984-0733410-4

Article copyright:
© Copyright 1984
American Mathematical Society