Piercing Quasi-Rectangles: On a Problem of Danzer and Rogers

It is an old problem of Danzer and Rogers to decide whether it is possible arrange O(1/epsilon) points in the unit square so that every rectangle of area epsilon 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 delta 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 delta. We show that the smallest number of points needed to pierce all quasi-rectangles of area e is Theta (1/epsilon log 1/epsilon).


Editor(s):
Dehne, F
Iacono, J
Sack, Jr
Published in:
Algorithms And Data Structures, 6844, 654-654
Presented at:
12th Algorithms and Data Structures Symposium (WADS 2011), Brooklyn, NY, AUG 15-17, 2011
Year:
2011
Publisher:
Berlin, Springer-Verlag Berlin
ISSN:
0302-9743
ISBN:
978-3-642-22299-3
Laboratories:




 Record created 2013-02-27, last modified 2018-03-17


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