Cutting families of convex sets

Katchalski, Meir; Lewis, Ted

doi:10.1090/S0002-9939-1980-0567992-9

Cutting families of convex sets
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by Meir Katchalski and Ted Lewis PDF

Proc. Amer. Math. Soc. 79 (1980), 457-461 Request permission

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 $|\mathcal {A}\backslash \mathcal {B}| \leqslant k$.

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