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On the Number of Directions Determined by the Common Tangents to a Family of Pairwise Disjoint Convex Sets in the Plane

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Abstract

Given a family \(F\) of \(n\) pairwise disjoint compact convex sets in the plane with non-empty interiors, let \(d(F)\) denote the number of directions determined by the set of lines which are tangent to two or more sets of \(F\). Let \(d_n\) denote the minimum value of \(d(F)\) as this parameter ranges over all such families of size \(n\). We prove that \(d_n\ge n-1\) for all \(n\) and show that this bound is tight for \(n=6\), and nearly tight for \(n=3,4,5\). The proof utilizes allowable interval sequences.

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Acknowledgments

Research for this paper was made possible thanks to support from the Israel Science Foundation and the European Research Council. This paper is dedicated to Ricky Pollack and Eli Goodman on the occasion of their 80th birthdays and in appreciation of their immense contributions to Discrete Geometry over the last several decades.

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  1. Department of Mathematics, Hebrew University, Jerusalem, Israel

    Mordechai Novick

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  1. Mordechai Novick
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Correspondence to Mordechai Novick.

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Novick, M. On the Number of Directions Determined by the Common Tangents to a Family of Pairwise Disjoint Convex Sets in the Plane. Discrete Comput Geom 53, 261–275 (2015). https://doi.org/10.1007/s00454-014-9654-x

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  • DOI: https://doi.org/10.1007/s00454-014-9654-x

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