Abstract
Let K denote an oval, a centrally symmetric compact convex domain with non-empty interior. A family of translates of K is said to have property T(k) if, for every subset of at most k translates, there exists a common line transversal intersecting all of them. Property T means that there is a line transversal to all the members of the family. Two translates, \(K_i\) and \(K_j\) of K, are said to be \(\varphi \)-disjoint, \(\varphi > 0\), if the concentric \(\varphi \)-enlarged copies of \(K_i\) and \( K_j\) are disjoint. It is known [5] that in a finite \(\varphi \)-disjoint family of translates of an oval, \(T(3) \Rightarrow T\) if \(\varphi >{2}\), and this bound is sharp. In this note, finite \(\varphi \)-disjoint T(4)-families of translates of an oval will be investigated.
Dedicated to Tudor Zamfirescu on his 70s birthday.
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Acknowledgments
The authors thank to I. Bárány, G. Ambrus, J. Eckhoff, and E. Roldán for the interesting discussions on line transversals to convex bodies.
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Heppes, A., Jerónimo-Castro, J. (2016). T(4) Families of \({\varphi }\)-Disjoint Ovals. In: Adiprasito, K., Bárány, I., Vilcu, C. (eds) Convexity and Discrete Geometry Including Graph Theory. Springer Proceedings in Mathematics & Statistics, vol 148. Springer, Cham. https://doi.org/10.1007/978-3-319-28186-5_13
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DOI: https://doi.org/10.1007/978-3-319-28186-5_13
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