On the discrepancy for boxes and polytopes (Q1295740)

The following problem is considered: Given an \(n\)-point set \(P\) in \(\mathbb{R}^d\), 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 \(P\cap B\) differs from the number of blue points in \(P\cap B\) by at most \(\Delta=O((\log n)^{d+1/2} \sqrt {\log\log n})\). The same asymptotic bound is shown for a similar problem where \(B\) is allowed to be any translated and scaled copy of a fixed convex polytope \(A\) in \(\mathbb{R}^d\), and such a bound also holds for the Lebesgue-measure discrepancy.