A simple proof of Stolarsky’s invariance principle

Abstract

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 L 2 \mathbb {L}_2 -discrepancy to give the distance integral of the uniform measure on the sphere which is a potential-theoretical quantity (Björck [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 L 2 \mathbb {L}_2 -discrepancy and vice versa. In this note we give a simple proof of the invariance principle using reproducing kernel Hilbert spaces.