  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{\"o}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.
