TIGHT UPPER-BOUNDS FOR THE DISCREPANCY OF HALF-SPACES

We show that the discrepancy of any n-point set P in the Euclidean d-space with respect to half-spaces is bounded by C(d)n(1/2-1/2d), that is, a mapping chi: P --> {-1,1} exists such that, for any half-space gamma, \Sigma(rho is an element of P boolean AND gamma) chi(rho)\less than or equal to C(d)n(1/2-1/2d). In fact, the result holds for arbitrary set systems as long as the primal shatter function is O(m(d)). This matches known lower bounds, improving previous upper bounds by a root log n factor.