Let S d be a unit sphere in ℝd+1, and let α be a positive real number. For pairwise different points x 1,x 2, . . . ,x N ∈ S d, we consider a functional E α (x 1,x 2, . . . ,x N ) = Σ i≠j ||x i − x j ||−α. The following theorem is proved: for α ≥ d − 2, the functional E α (x 1,x 2, . . . ,x N ) does not have local maxima.
References
A. Bondarenko and M. Viazovska, “Spherical designs via Brouwer fixed point theorem,” SIAM J. Discrete Math., 24, 207–217 (2010).
H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres,” J. Amer. Math. Soc., 20, No. 1, 99–148 (2007).
E. B. Saff and A. B. J. Kuijlaars, “Distributing many points on a sphere,” Math. Intelligencer, 19, No. 1, 5–11 (1997).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 10, pp. 1427–1429, October, 2013.
Rights and permissions
About this article
Cite this article
Radchenko, D.V. Local Maxima of the Potential Energy on Spheres. Ukr Math J 65, 1585–1587 (2014). https://doi.org/10.1007/s11253-014-0880-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-014-0880-4