Let $$C_1, \dots , C_n$$ C 1 , ⋯ , C n denote the 1 / n-neighborhood of n great circles on $$\mathbb {S}^2$$ S 2 . We are interested in how much these areas have to overlap and prove the sharp bounds $$\begin{aligned} \mathop {\mathop {\sum }\limits _{i, j = 1}}\limits _ {i \ne j}^{n}{|C_i \cap C_j|^s} \gtrsim _s {\left\{ \begin{array}{ll} n^{2 - 2s} \qquad &{}\text{ if }~0 \le s < 2, \\ n^{-2} \log {n} \qquad &{}\text{ if }~s = 2,\\ n^{1- 3s/2} \qquad &{}\text{ if }~s > 2. \end{array}\right. } \end{aligned}$$ ∑ i , j = 1 i ≠ j n | C i ∩ C j | s ≳ s n 2 - 2 s if 0 ≤ s < 2 , n - 2 log n if s = 2 , n 1 - 3 s / 2 if s > 2 . For $$s=1$$ s = 1 there are arrangements for which the sum of mutual overlap is uniformly bounded (for the analogous problem in $$\mathbb {R}^2$$ R 2 this is impossible, the lower bound is $$\gtrsim \log {n}$$ ≳ log n ). There are strong connections to minimal energy configurations of n charged electrons on $$\mathbb {S}^2$$ S 2 (the Thomson problem).
Discrete & Computational Geometry – Springer Journals
Published: Apr 11, 2018
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