A Variant of the Hadwiger–Debrunner (p, q)-Problem in the Plane

Sathish Govindarajan; Gabriel Nivasch

doi:10.1007/s00454-015-9723-9

A Variant of the Hadwiger–Debrunner (p, q)-Problem in the Plane

Sathish Govindarajan, Gabriel Nivasch

Department of Computer Science

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

Let X be a convex curve in the plane (say, the unit circle), and let S be a family of planar convex bodies such that every two of them meet at a point of X. Then S has a transversal (Formula presented.) of size at most 1.75×10⁹. Suppose instead that S only satisfies the following “(p, 2)-condition”: Among every p elements of S, there are two that meet at a common point of X. Then S has a transversal of size O(p⁸). For comparison, the best known bound for the Hadwiger–Debrunner (p, q)-problem in the plane, with q=3, is O(p⁶). Our result generalizes appropriately for (Formula presented.) if (Formula presented.) is, for example, the moment curve.

Original language	English
Pages (from-to)	637-646
Number of pages	10
Journal	Discrete and Computational Geometry
Volume	54
Issue number	3
DOIs	https://doi.org/10.1007/s00454-015-9723-9
State	Published - 7 Oct 2015

Keywords

Convex set
Fractional Helly
Hadwiger–Debrunner (p, q)-problem
Helly’s theorem
Transversal
Weak epsilon-net

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10.1007/s00454-015-9723-9

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abstract = "Let X be a convex curve in the plane (say, the unit circle), and let S be a family of planar convex bodies such that every two of them meet at a point of X. Then S has a transversal (Formula presented.) of size at most 1.75×109. Suppose instead that S only satisfies the following “(p, 2)-condition”: Among every p elements of S, there are two that meet at a common point of X. Then S has a transversal of size O(p8). For comparison, the best known bound for the Hadwiger–Debrunner (p, q)-problem in the plane, with q=3, is O(p6). Our result generalizes appropriately for (Formula presented.) if (Formula presented.) is, for example, the moment curve.",

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KW - Helly’s theorem

KW - Transversal

KW - Weak epsilon-net

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A Variant of the Hadwiger–Debrunner (p, q)-Problem in the Plane

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Keywords

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