Spherical Discrepancy Minimization and Algorithmic Lower Bounds for Covering the Sphere

Inspired by the boolean discrepancy problem, we study the following optimization problem which we term Spherical Discrepancy: given m unit vectors v 1 , . . . , v m , find another unit vector x that minimizes max i x, v i . We show that Spherical Discrepancy is APX-hard and develop a multiplicative weights-based algorithm that achieves optimal worst-case error bounds up to lower order terms. We use our algorithm to give the first non-trivial lower bounds for the problem of covering a hypersphere by hyperspherical caps of uniform volume at least 2 -o( √ n) . We accomplish this by proving a related covering bound in Gaussian space and showing that in this large cap regime the bound transfers to spherical space. Up to a log factor, our lower bounds match known upper bounds in the large cap regime.