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

Well-Distributed Great Circles on $$\mathbb {S}^2$$ S 2 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). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete & Computational Geometry Springer Journals

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

, Volume 60 (1) – Apr 11, 2018
17 pages

/lp/springer_journal/well-distributed-great-circles-on-mathbb-s-2-s-2-Yht01FSOBY
Publisher
Springer Journals
Subject
Mathematics; Combinatorics; Computational Mathematics and Numerical Analysis
ISSN
0179-5376
eISSN
1432-0444
D.O.I.
10.1007/s00454-018-9994-z
Publisher site
See Article on Publisher Site

### Abstract

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).

### Journal

Discrete & Computational GeometrySpringer Journals

Published: Apr 11, 2018

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