TY - JOUR

T1 - A Variant of the Hadwiger–Debrunner (p, q)-Problem in the Plane

AU - Govindarajan, Sathish

AU - Nivasch, Gabriel

N1 - Publisher Copyright:
© 2015, Springer Science+Business Media New York.

PY - 2015/10/7

Y1 - 2015/10/7

N2 - 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.

AB - 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.

KW - Convex set

KW - Fractional Helly

KW - Hadwiger–Debrunner (p, q)-problem

KW - Helly’s theorem

KW - Transversal

KW - Weak epsilon-net

UR - http://www.scopus.com/inward/record.url?scp=84941022223&partnerID=8YFLogxK

U2 - 10.1007/s00454-015-9723-9

DO - 10.1007/s00454-015-9723-9

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AN - SCOPUS:84941022223

SN - 0179-5376

VL - 54

SP - 637

EP - 646

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 3

ER -