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Geometric Set Systems

  • Conference paper
European Congress of Mathematics

Part of the book series: Progress in Mathematics ((PM,volume 169))

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.

Part of this survey was written while the author was visiting ETH Zürich, whose support is gratefully acknowledged. Also supported by Czech Republic Grant GAČR 0194/1996 and by Charles University grants No. 193,194/1996.

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  1. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jiří MatouŠek

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  1. Hungarian Academy of Sciences, Mathematical Institute, Reáltanoda str. 13-15, H-1053, Budapest, Hungary

    A. Balog , G. O. H. Katona  & D. Sza’sz ,  & 

  2. Mathematical Institute, Technical University of Budapest, H-1521, Budapest, Hungary

    A. Recski

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MatouŠek, J. (1998). Geometric Set Systems. In: Balog, A., Katona, G.O.H., Recski, A., Sza’sz, D. (eds) European Congress of Mathematics. Progress in Mathematics, vol 169. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8898-1_1

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