Tight upper bounds for the discrepancy of half-spaces |
| |
Authors: | J Matou?ek |
| |
Institution: | (1) Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic |
| |
Abstract: | We show that the discrepancy of anyn-point setP in the Euclideand-space with respect to half-spaces is bounded byC
d
n
1/2−1/2d
, that is, a mapping χ:P→{−1,1} exists such that, for any half-space γ, γ, |Σ
p∈P⋂γ
χ(p)|≤C
d
n
1/2-1/2d
. In fact, the result holds for arbitrary set systems as long as theprimal shatter function isO(m
d
). This matches known lower bounds, improving previous upper bounds by a
factor.
Part of this research was done at the Third Israeli Computational Geometry Workshop in Ramot and during a visit at Tel Aviv
University in December 1993. Also supported in part by Charles University Grant No. 351 and Czech Republic Grant GAČR 201/93/2167. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|