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TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
Geometric Set Systems
, 1998
"... 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 higherdimensional spaces by other classes of simple geometric figures (simplices, balls, ellipsoids, e ..."
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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 higherdimensional 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 VapnikChervonenkis 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 described tools might be useful in other areas of mathematics too. 1 Introduction For a set system S ` 2 X on an arbitrary ground set X and for A ` X, we write Sj A = fS " A; S 2 Sg for the set system induced by S on A (or the trace of S on A). Let H den...
Discrepancy of Point Sequences on Fractal Sets
"... We consider asymptotic bounds for the discrepancy of point sets on a class of fractal sets. By a method of R. Alexander, we prove that for a wide class of fractals, the L 2 discrepancy (and consequently also the worstcase discrepancy) of an Npoint set with respect to halfspaces is at least of the ..."
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We consider asymptotic bounds for the discrepancy of point sets on a class of fractal sets. By a method of R. Alexander, we prove that for a wide class of fractals, the L 2 discrepancy (and consequently also the worstcase discrepancy) of an Npoint set with respect to halfspaces is at least of the order N^(1/21/(2s)) , where s is the Hausdorff dimension of the fractal. We also show that for many fractals, this bound is tight for the L 2 discrepancy. Determining the correct order of magnitude of the worstcase discrepancy remains a challenging open problem.