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Discrete & Computational Geometry

, Volume 60, Issue 1, pp 40–56 | Cite as

Well-Distributed Great Circles on \(\mathbb {S}^2\)

  • Stefan Steinerberger
Article
  • 105 Downloads

Abstract

Let \(C_1, \dots , C_n\) denote the 1 / n-neighborhood of n great circles on \(\mathbb {S}^2\). We are interested in how much these areas have to overlap and prove the sharp bounds
$$\begin{aligned} \mathop {\mathop {\sum }\limits _{i, j = 1}}\limits _ {i \ne j}^{n}{|C_i \cap C_j|^s} \gtrsim _s {\left\{ \begin{array}{ll} n^{2 - 2s} \qquad &{}\text{ if }~0 \le s < 2, \\ n^{-2} \log {n} \qquad &{}\text{ if }~s = 2,\\ n^{1- 3s/2} \qquad &{}\text{ if }~s > 2. \end{array}\right. } \end{aligned}$$
For \(s=1\) there are arrangements for which the sum of mutual overlap is uniformly bounded (for the analogous problem in \(\mathbb {R}^2\) this is impossible, the lower bound is \(\gtrsim \log {n}\)). There are strong connections to minimal energy configurations of n charged electrons on \(\mathbb {S}^2\) (the Thomson problem).

Keywords

Intersection of great circles Minimal overlap Riesz energy Thomson problem 

Mathematics Subject Classification

31A15 52C35 (Primary) 52C10 (Secondary) 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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