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

Abstract

Weyl's discrepancy measure induces a norm on R n 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. MotivationIn the mathematical literature discrepancy theory is devoted to problems related to irregularities of distributions. In this context the term discrepancy refers to a measure that evaluates to which extent a given distribution deviates from total uniformity in measure-theoretic, combinatorial and geometric settings. This theory goes back to Weyl [39] and is still an active field of research, see, e.g., [3,12,19]. Applications can be found in the field of numerical integration, especially for Monte Carlo methods in high dimensions, see, e.g., [28,36,40], or in computational geometry, see, e.g., [1,9,21]. For applications to data storage problems on parallel disks, see [10,13] and for halftoning images, see [31].This paper is motivated by [24] which applies Weyl's discrepancy concept in order to derive an ordering-dependent norm for measuring the (dis-)similarity between patterns. In this context the focus lies on evaluating the auto-misalignment that measures

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