Showing papers by "Jiří Matoušek published in 1998"
TL;DR: It is pointed out that if the number of points is not large enough in terms of the dimension then nearly the lowest possible L 2 -discrepancy is attained by a pathological point set, and hence the L 1 -Discrepancy may not be very relevant for relatively small sets.
290 citations
01 Jan 1998
TL;DR: This work considers the set system on X whose sets are all intersections of X with a halfplane, and focuses on applications in discrepancy theory, in combinatorial geometry, in derandomization of geometric algorithms, and in geometric range searching.
Abstract: Let X be a finite point set in the plane. We consider the set system on X whose sets are all intersections of X with a halfplane. Similarly one can investigate set systems defined on point sets in higher-dimensional spaces by other classes of simple geometric figures (simplices, balls, ellipsoids, etc.). It turns out that simple combinatorial properties of such set systems (most notably the Vapnik-Chervonenkis dimension and related concepts of shatter functions) play an important role in several areas of mathematics and theoretical computer science. Here we concentrate on applications in discrepancy theory, in combinatorial geometry, in derandomization of geometric algorithms, and in geometric range searching. We believe that the tools described might be useful in other areas of mathematics too.
20 citations
TL;DR: Wasilkowski and Woźniakowski proved that p *⩽1.4779 was proved, by combining known bounds for the error of numerical integration and using their relation to L 2 -discrepancy.
13 citations