Geometric Characterization of Weyl’s Discrepancy Norm in Terms of Its n-Dimensional Unit Balls

Abstract

Weyl’s discrepancy measure induces a norm on ℝn which shows a monotonicity and a Lipschitz property when applied to differences of index-shifted sequences. It turns out that its n-dimensional unit ball is a zonotope that results from a multiple sheared projection from the (n+1)-dimensional hypercube which can be interpreted as a discrete differentiation. This characterization reveals that this norm is the canonical metric between sequences of differences of values from the unit interval in the sense that the n-dimensional unit ball of the discrepancy norm equals the space of such sequences.