The Stolarsky Principle and Energy Optimization on the Sphere


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. 1.

    Aigner, M., Ziegler, G.: Proofs from The Book, 5th edn. Springer, Berlin (2014)

    MATH  Google Scholar 

  2. 2.

    Beck, J.: Some upper bounds in the theory of irregularities of distribution. Acta Arith. 43(2), 115–130 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Beck, J.: Sums of distances between points on a sphere-an application of the theory of irregularities of distribution to discrete geometry. Mathematika 31(1), 33–41 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Benedetto, J., Fickus, M.: Finite normalized tight frames. Adv. Comput. Math. 18(2–4), 357–385 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bilyk, D., Dai, F.: Geodesic distance Riesz energy on the sphere, 2017, to appear, available at arXiv: 1612.08442

  6. 6.

    Bilyk, D., Lacey, M.: Random tessellations, restricted isometric embeddings, and one bit sensing, 2017, to appear, available at arXiv:1512.06697

  7. 7.

    Bilyk, D., Lacey, M.: One bit sensing, discrepancy, and Stolarsky principle. Sb. Math. 208(6), 744–763 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Björck, G.: Distributions of positive mass, which maximize a certain generalized energy integral. Ark. Mat. 3, 255–269 (1956)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Blümlinger, M.: Slice discrepancy and irregularities of distribution on spheres. Mathematika 38(1), 105–116 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Borodachov, S., Hardin, D., Saff, E.: Minimal Discrete Energy on Rectifiable Sets, Springer, Monographs in Math. (to appear)

  11. 11.

    Brauchart, J.: About the second term of the asymptotics for optimal Riesz energy on the sphere in the potential-theoretical case. Integral Transforms Spec. Funct. 17(5), 321–328 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Brauchart, J., Hardin, D., Saff, E.: Discrete energy asymptotics on a Riemannian circle. Unif. Distrib. Theory 7(2), 77–108 (2012)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Brauchart, J.S., Dick, J.: A simple proof of Stolarsky’s invariance principle. Proc. Am. Math. Soc. 141, 2085–2096 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Drmota, M., Tichy, R.: Sequences, Discrepancies and Applications. Lecture Notes in Mathematics, vol. 1651. Springer, Berlin (1997)

    MATH  Google Scholar 

  15. 15.

    Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics. Springer, New York (2013)

    Book  MATH  Google Scholar 

  16. 16.

    Damelin, S., Hickernell, F., Ragozin, D., Zeng, X.: On energy, discrepancy and group invariant measures on measurable subsets of Euclidean space. J. Fourier Anal. Appl. 16(6), 813–839 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Fejes Tóth, L.: On the sum of distances determined by a pointset. Acta Math. Acad. Sci. Hungar. 7, 397–401 (1956)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Fejes Tóth, L.: Über eine Punktverteilung auf der Kugel (German). Acta Math. Acad. Sci. Hung. 10, 13–19 (1959)

    Article  MATH  Google Scholar 

  19. 19.

    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press Inc, San Diego, CA (2000)

    MATH  Google Scholar 

  20. 20.

    Groemer, H.: Geometric Applications of Fourier Series and Spherical Harmonics. Encyclopedia of Mathematics and Its Applications, vol. 61. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  21. 21.

    He, H., Basu, K., Zhao, Q., Owen, A.: Permutation \(p\)-value approximation via generalized Stolarsky invariance, to appear, available at arXiv:1603.02757

  22. 22.

    Hinrichs, A., Nickolas, P., Wolf, R.: A note on the metric geometry of the unit ball. Math. Z. 268(3–4), 887–896 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Jiang, M.: On the sum of distances along a circle. Discrete Math. 308(10), 2038–2045 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Kuijlaars, A., Saff, E.: Asymptotics for minimal discrete energy on the sphere. Trans. Am. Math. Soc. 350(2), 523–538 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Larcher, H.: Solution of a geometric problem by Fejes Tóth. Mich. Math. J. 9, 45–51 (1962)

    Article  MATH  Google Scholar 

  26. 26.

    Matoušek, J.: Geometric Discrepancy. An Illustrated Guide. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  27. 27.

    Nielsen, F.: On the sum of the distances between \(n\) points on a sphere. Nordisk Mat. Tidskr. 13, 45–50 (1965). (Danish)

    MATH  Google Scholar 

  28. 28.

    Plan, Y., Vershynin, R.: Dimension reduction by random uniform tessellation. Discrete Comput. Geom. to appear

  29. 29.

    Schoenberg, I.: Some extremal problems for positive definite sequences and related extremal convex conformal maps of the circle. Nederl. Akad. Wetensch. Proc. Ser A 20, 28–37 (1958)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Skriganov, M.: Point distributions in compact metric spaces, to appear, available at arXiv:1512.00364

  31. 31.

    Sperling, G.: Lösung einer elementargeometrischen Frage von Fejes Tóth. (German). Arch. Math. 11, 69–71 (1960)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Stolarsky, K.B.: Sums of distances between points on a sphere. II. Proc. Am. Math. Soc. 41, 575–582 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Stolarsky, K.B.: Spherical distributions of \(N\) points with maximal distance sums are well spaced. Proc. Am. Math. Soc. 48, 203–206 (1975)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Torquato, S.: Reformulation of the covering and quantizer problems as ground states of interacting particles. Phys. Rev. E 82(5), 1–52 (2010)

  35. 35.

    Wagner, G.: On means of distances on the surface of a sphere. II. Upper bounds. Pac. J. Math. 154(2), 381–396 (1992)

    Article  MATH  Google Scholar 

  36. 36.

    Woźniakowski, H.: Average case complexity of multivariate integration. Bull. Am. Math. Soc. 24(1), 185–194 (1991)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Dmitriy Bilyk.

Additional information

This work was partially supported by the Simons Foundation Collaboration Grant (Bilyk), NSERC Canada under Grant RGPIN 04702 (Dai), and the NSF Graduate Research Fellowship 00039202 (Matzke). The first two authors are grateful to CRM Barcelona: this collaboration originated during their participation in the research program “Constructive Approximation and Harmonic Analysis” (Bilyk’s trip was sponsored by NSF Grant DMS 1613790).

Communicated by Doug Hardin.

Appendix: Mean-Square Geodesic Distance

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

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


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

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 \(I (\mu ) = \frac{1}{2}\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bilyk, D., Dai, F. & Matzke, R. The Stolarsky Principle and Energy Optimization on the Sphere. Constr Approx 48, 31–60 (2018).

Download citation


  • Discrepancy
  • Energy minimization
  • Stolarsky principle

Mathematics Subject Classification

  • 11K38
  • 74G65
  • 42A82