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From the Separable Tammes Problem to Extremal Distributions of Great Circles in the Unit Sphere
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From the Separable Tammes Problem to Extremal Distributions of Great Circles in the Unit Sphere
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Journal Article
From the Separable Tammes Problem to Extremal Distributions of Great Circles in the Unit Sphere
2024
Overview
A family of spherical caps of the 2-dimensional unit sphere S2 is called a totally separable packing in short, a TS-packing if any two spherical caps can be separated by a great circle which is disjoint from the interior of each spherical cap in the packing. The separable Tammes problem asks for the largest density of given number of congruent spherical caps forming a TS-packing in S2. We solve this problem up to eight spherical caps and upper bound the density of any TS-packing of congruent spherical caps in terms of their angular radius. Based on this, we show that the centered separable kissing number of unit balls in Euclidean 3-space is 8. Furthermore, we prove bounds for the maximum of the smallest inradius of the cells of the tilings generated by n>1 great circles in S2. Next, we prove dual bounds for TS-coverings of S2 by congruent spherical caps. Here a covering of S2 by spherical caps is called a totally separable covering in short, a TS-covering if there exists a tiling generated by finitely many great circles of S2 such that the cells of the tiling are covered by pairwise distinct spherical caps of the covering. Finally, we extend some of our bounds on TS-coverings to spherical spaces of dimension >2.
Publisher
Springer Nature B.V
Subject
