A simple Proof of Stolarsky's Invariance Principle

Stolarsky [Proc. Amer. Math. Soc. 41 (1973), 575--582] showed a beautiful relation that balances the sums of distances of points on the unit sphere and their spherical cap $\mathbb{L}_2$-discrepancy to give the distance integral of the uniform measure on the sphere a potential-theoretical quantity (Bj{ö}rck [Ark. Mat. 3 (1956), 255--269]). Read differently it expresses the worst-case numerical integration error for functions from the unit ball in a certain Hilbert space setting in terms of the $\mathbb{L}_2$-discrepancy and vice versa (first author and Womersley [Preprint]). In this note we give a simple proof of the invariance principle using reproducing kernel Hilbert spaces.