On the Discrepancy for Boxes and Polytopes

Abstract

We consider so-called Tusnady’s problem in dimension d: Given an n-point set P in R d , color the points of P red or blue in such a way that for any d-dimensional interval B, the number of red points in \(\) differs from the number of blue points in \(\) by at most Δ, where \(\) should be as small as possible. We slightly improve previous results of Beck, Bohus, and Srinivasan by showing that \(\), with a simple proof. The same asymptotic bound is shown for an analogous problem where B is allowed to be any translated and scaled copy of a fixed convex polytope A in R d . Here the constant of proportionality depends on A and we give an explicit estimate. The same asymptotic bounds also follow for the Lebesgue-measure discrepancy, which improves and simplifies results of Beck and of Karolyi.