AN ELEMENTARY APPROACH TO LOWER BOUNDS IN GEOMETRIC DISCREPANCY

For each d greater than or equal to 2, it is possible to place n points in d-space so that, given any two-coloring of the points, a half-space exists within which one color outnumbers the other by as much as Cn(1/2-1/2d), for some constant c > 0 depending on d. This result was proven in a slightly weaker form by Beck and the bound was later tightened by Alexander. It was recently shown to be asymptotically optimal by MatouSek. We present a proof of the lower bound, which is based on Alexander's technique but is technically simpler and more accessible. We present three variants of the proof, for three different cases, to provide more intuitive insight into the ''large-discrepancy'' phenomenon. We also give geometric and probabilistic interpretations of the technique.