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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.

  

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

 

Collections: Mathematics