Combinatorial Discrepancy for Boxes via the Ellipsoid-Infinity Norm

The ellipsoid-infinity norm of a real m × n matrix A, denoted by ‖A‖E∞, is the minimum `∞ norm of an 0-centered ellipsoid E ⊆ R that contains all column vectors of A. This quantity, introduced by the second author and Talwar in 2013, is polynomial-time computable and approximates the hereditary discrepancy herdiscA as follows: ‖A‖E∞/O(logm) ≤ herdiscA ≤ ‖A‖E∞ ·O( √ logm). Here we show that both of the inequalities are asymptotically tight in the worst case, and we provide a simplified proof of the first inequality. We establish a number of favorable properties of ‖.‖E∞, such as the triangle inequality and multiplicativity w.r.t. the Kronecker (or tensor) product. We then demonstrate on several examples the power of the ellipsoidinfinity norm as a tool for proving lower and upper bounds in discrepancy theory. Most notably, we prove a new lower bound of Ω(logd−1 n) for the d-dimensional Tusnády problem, asking for the combinatorial discrepancy of an n-point set in R with respect to axis-parallel boxes. For d > 2, this improves the previous best lower bound, which was of order approximately log(d−1)/2 n, and it comes close to the best known upper bound of O(log n), for which we also obtain a new, very simple proof.

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