The Stolarsky Principle and Energy Optimization on the Sphere

Abstract

The classical Stolarsky invariance principle connects the spherical cap $L^2$ discrepancy of a finite point set on the sphere to the pairwise sum of Euclidean distances between the points. In this paper, we further explore and extend this phenomenon. In addition to a new elementary proof of this fact, we establish several new analogs, which relate various notions of discrepancy to different discrete energies. In particular, we find that the hemisphere discrepancy is related to the sum of geodesic distances. We also extend these results to arbitrary measures on the sphere and arbitrary notions of discrepancy and apply them to problems of energy optimization and combinatorial geometry and find that, surprisingly, the geodesic distance energy behaves differently than its Euclidean counterpart.

Appendix: Mean-Square Geodesic Distance

The following integral arises in the formulations of Stolarsky principles for wedges (6.1) and slices (6.3):

$$\begin{aligned} V_d = \int \limits _{\mathbb S^d} \!\int \limits _{\mathbb S^d} \big ( d(x,y) \big )^2 \,\, d\sigma (x) \, d\sigma (y); \end{aligned}$$

hence we compute it and examine its properties. A standard calculation yields that

$$\begin{aligned} V_d = \frac{1}{\pi ^2} \cdot \frac{\omega _{d-1}}{\omega _d} \int _{0}^\pi \phi ^2 \sin ^{d-1} \phi \, d\phi . \end{aligned}$$

Applying the recursive relation [19]

$$\begin{aligned} \int x^m \sin ^n x \, dx&= \frac{x^{m-1} \sin ^{n-1} x}{n^2 } \, \big (m \sin x - nx \cos x \big ) + \\&\quad +\, \frac{n-1}{n} \int x^m \sin ^{n-2} x \,dx - \frac{m(m-1)}{n^2} \int x^{m-2} \sin ^n x \, dx \end{aligned}$$

with $m=2$ and $n=d-1$, as well as the facts that

$$\begin{aligned} \frac{\omega _{d-1}}{\omega _d} = \frac{d-1}{d-2} \cdot \frac{\omega _{d-3}}{\omega _{d-2}} \end{aligned}$$

and

$$\begin{aligned} \int _0^\pi \sin ^{d-1} \phi \, d\phi = \frac{\sqrt{\pi } \Gamma (d/2)}{\Gamma \big ( (d+1)/2\big )}, \end{aligned}$$

we obtain the recursive relation

$$\begin{aligned} V_d = V_{d-2} - \frac{2}{\pi ^2 (d-1)^2}. \end{aligned}$$

Together with simple identities $V_1 = \frac{1}{3}$ and $V_2 = \frac{1}{2} - \frac{2}{\pi ^2}$ (or even $V_0 = \frac{1}{2}$), this yields:

Lemma 7.1

For odd values of $d\ge 1$,

$$\begin{aligned} V_d = \frac{1}{3} - \frac{2}{\pi ^2} \sum _{k=1}^{{(d-1)}/{2}} \frac{1}{(2k)^2 }, \end{aligned}$$

while for even values of $d\ge 2$,

$$\begin{aligned} V_d = \frac{1}{2} - \frac{2}{\pi ^2} \sum _{k=1}^{{d}/{2}} \frac{1}{(2k-1)^2 }. \end{aligned}$$

Since $\displaystyle {\sum _{k=1}^\infty \frac{1}{(2k)^2} = \frac{\pi ^2}{24}}$ and $\displaystyle {\sum _{k=1}^\infty \frac{1}{(2k-1)^2} = \frac{\pi ^2}{8}}$, we find that

$$\begin{aligned} \displaystyle {\lim _{d\rightarrow \infty } V_d = \frac{1}{4}}, \end{aligned}$$

which is consistent with the concentration of measure phenomenon (“most points” on the high-dimensional sphere are nearly orthogonal).

Notice that this confirms the result of Theorem 4.7 that, unless $d=0$, the uniform distribution $\sigma$ is not a maximizer of $I (\mu ) = \int \limits _{\mathbb S^d} \!\int \limits _{\mathbb S^d} \big ( d(x,y) \big )^2 \,\, d\mu (x) \, d\mu (y)$, since for $\mu = \frac{1}{2} {\delta _p }+ \frac{1}{2} {\delta _{-p}}$ we have $I (\mu ) = \frac{1}{2}$.