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 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...

We study the problem of finding optimal linear decision trees for classifying a set of points in IR d partitioned into concept classes, where d is a fixed, but arbitrary, constant. We show that optimal decision tree construction is NP-complete, even for 3-dimensional point sets. Nevertheless, we can prove a number of interesting approximation bounds on the use of random sampling for finding optimal splitting hyperplanes in greedy decision tree constructions. We give experimental evidence that, while providing asymptotic guarantees on split quality, this random sampling approach behaves as good in practice as uniform randomization strategies that do not provide such guarantees. Finally, we provide experimental justification for coupling this random sampling strategy with locally-greedy "hill climbing" methods. 1 Introduction A general framework for machine learning is that one is given a (hopefully representative) sample S of n points taken from some much larger (possibly infinite) ...