Piercing quasi-rectangles-On a problem of Danzer and Rogers

Authors:
JáNos Pach;GáBor Tardos
Affiliations:
EPFL, Lausanne, Switzerland and Rényi Institute, Budapest, Hungary;Rényi Institute, Budapest, Hungary and Department of Computer Science, Simon Fraser University, Burnaby, Canada
Venue:
Journal of Combinatorial Theory Series A
Year:
2012

Citing 5
Cited 0

Some new bounds for Epsilon-nets

SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Almost tight bounds for &egr;-nets

Discrete & Computational Geometry
Weak ε-nets and interval chains

Journal of the ACM (JACM)
A Non-linear Lower Bound for Planar Epsilon-Nets

FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Tight lower bounds for the size of epsilon-nets

Proceedings of the twenty-seventh annual symposium on Computational geometry

Quantified Score

Hi-index	0.00

Visualization

Abstract

It is an old problem of Danzer and Rogers to decide whether it is possible to arrange O(1@e) points in the unit square so that every rectangle of area @e0 within the unit square contains at least one of them. We show that the answer to this question is in the negative if we slightly relax the notion of rectangles, as follows. Let @d be a fixed small positive number. A quasi-rectangle is a region swept out by a continuously moving segment s, with no rotation, so that throughout the motion the angle between the trajectory of the center of s and its normal vector remains at most @d. We show that the smallest number of points needed to pierce all quasi-rectangles of area @e0 within the unit square is @Q(1@elog1@e).