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A non-linear lower bound for planar epsilon-nets We show that the minimum possible size of an -net for point objects and line (or rectangle)-
 

Summary: A non-linear lower bound for planar epsilon-nets
Noga Alon
Abstract
We show that the minimum possible size of an -net for point objects and line (or rectangle)-
ranges in the plane is (slightly) bigger than linear in 1/ . This settles a problem raised by Matousek,
Seidel and Welzl in 1990.
1 Introduction
A range space S is a pair (X, R), where X is a (finite or infinite) set and R is a (finite or infinite)
family of subsets of X. The members of X are called points and those of R are called ranges. If A is a
subset of X then PR(A) = {rA : r R} is the projection of R on A. In case this projection contains
all subsets of A we say that A is shattered. The Vapnik-Chervonenkis dimension (or VC-dimension) of
S, denoted by V C(S), is the maximum cardinality of a shattered subset of X. If there are arbitrarily
large shattered subsets then V C(S) = .
For a finite set of points A in a range space, a subset N A is an -net for A if any range
r R satisfying |r A| |A| contains at least one point of N. The theory of VC-dimension
and -nets has played a central role in discrete and computational geometry, and has been used
in a variety of applications including range searching, geometric partitions, and bounds on various
incidence problems, as well as in other mathematical areas such as statistics, computational learning,
discrepancy theory and combinatorics.
A well known result of Haussler and Welzl [19], following earlier work of Vapnik and Chervonenkis

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics