Tight upper bounds for the discrepancy of half-spaces

Abstract

We show that the discrepancy of any n-point set P in the Euclidean d-space with respect to half-spaces is bounded by Cd n1/2-1/2 d, that is, a mapping χ:P→{-1,1} exists such that, for any half-space γ, γ, |Σp∈P{n-ary intersection}γ χ(p)|≤Cd n1/2-1/2 d . In fact, the result holds for arbitrary set systems as long as the primal shatter function is O(md ). This matches known lower bounds, improving previous upper bounds by a {Mathematical expression} factor. © 1995 Springer-Verlag New York Inc.