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

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 $$\begin{aligned} \mathop {\mathop {\sum }\limits _{i, j = 1}}\limits _ {i \ne j}^{n}{|C_i \cap C_j|^s} \gtrsim _s {\left\{ \begin{array}{ll} n^{2 - 2s} \qquad &{}\text{ if }~0 \le s 2. \end{array}\right. } \end{aligned}$$ 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).