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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Cutting families of convex sets


Authors: Meir Katchalski and Ted Lewis
Journal: Proc. Amer. Math. Soc. 79 (1980), 457-461
MSC: Primary 52A35; Secondary 52A10
MathSciNet review: 567992
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Abstract: A family of convex sets in the plane admits a common transversal if there is a straight line which intersects (cuts) each member of the family.

It is shown that there is a positive integer k such that for any compact convex set C in the plane and for any finite family $ \mathcal{A}$ of pairwise disjoint translates of C: If each 3-membered subfamily of $ \mathcal{A}$ admits a common transversal then there is a subfamily $ \mathcal{B}$ of $ \mathcal{A}$ such that $ \mathcal{B}$ admits a common transversal and $ \vert\mathcal{A}\backslash \mathcal{B}\vert \leqslant k$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0567992-9
PII: S 0002-9939(1980)0567992-9
Keywords: Common transversal, convex sets
Article copyright: © Copyright 1980 American Mathematical Society