Quasi-Orthogonality via Finite-Di erencing: An Elementary Approach to Geometric Discrepancy

It is possible to place n points in d-space so that, given any two-coloring of the points, there exists a halfspace within which one color dominates the other by as much as cn 1=2d, for some constant c > 0. This result was proven in a slightly weaker form by Beck and the bound was later tightened by Alexander. It was shown to be quasi-optimal by Matou sek, Welzl, and Wernisch. The lower bound proofs are highly technical and do not provide much intuitive insight into the \large-discrepancy" phenomenon. We develop a proof technique which allows us to rederive the same lower bound in a much simpler fashion. We give a probabilistic interpretation of the result and we discuss the connection of our method to Beck's Fourier transform approach. We also provide a quasi-optimal lower bound on the discrepancy of xed-size rotated boxes, which signi cantly improves the previous bound.