Abstract
A cap of spherical radius α on a unit d-sphere S is the set of points within spherical distance α from a given point on the sphere. Let \({\cal F}\) be a finite set of caps lying on S. We prove that if no hyperplane through the center of S divides \({\cal F}\) into two non-empty subsets without intersecting any cap in \({\cal F}\), then there is a cap of radius equal to the sum of radii of all caps in \({\cal F}\) covering all caps of \({\cal F}\) provided that the sum of radii is less than π/2.
This is the spherical analog of the so-called Circle Covering Theorem by Goodman and Goodman and the strengthening of Fejes Tóth’s zone conjecture proved by Jiang and the author.
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Acknowledgements
The author thanks Alexey Balitskiy for the fruitful discussions of his work [3]. Besides, the author thanks the anonymous referees whose remarks helped fix some errors and improve the presentation.
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The author was supported by the Russian Federation Government through Grant No. 075-15-2019-1926. The author is a Young Russian Mathematics award winner and would like to thank its sponsors and jury.
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Polyanskii, A. A Cap Covering Theorem. Combinatorica 41, 695–702 (2021). https://doi.org/10.1007/s00493-021-4554-1
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DOI: https://doi.org/10.1007/s00493-021-4554-1