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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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Keywords: Common transversal, convex sets
Article copyright: © Copyright 1980 American Mathematical Society