Abstract
In this paper, we consider the problem on the contact number of a sphere in a three-dimensional hyperbolic space and a three-dimensional spherical space. This problem is equivalent to the Tammes problem on the maximal value of minimums of spherical distances for an N-point set on the unit sphere of the Euclidean space.
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References
K. Anstreicher, “The thirteen spheres: A new proof,” Discr. Comput. Geom., 31, 613–625 (2004).
C. Bachoc and F. Vallentin, “New upper bounds for kissing numbers from semidefinite programming,” Discr. Comput. Geom., 21, 909–924 (2008).
K. Böröczky, “The problem of Tammes for n = 11,” Stud. Sci. Math. Hungar., 18, 165–171 (1983).
K. Böröczky, “The Newton–Gregory problem revisited,” in: Discrete Geometry (A. Bezdek, ed.), Marcel Dekker (2003), pp. 103–110.
K. Böröczky and L. Szabó, “Arrangements of 13 points on a sphere,” in: Discrete Geometry (A. Bezdek, ed.), Marcel Dekker (2003), pp. 111–184.
P. Boyvalenkov, S. Dodunekov, and O. R.Musin, “A survey on the kissing number,” Serd. Math. J., 38, 507–522 (2012).
L. Danzer, Endliche Punktmengen auf der 2-Sphäre mit möglichst grossen Minimalabstand, Universit ät Göttingen (1963).
L. Danzer, “Finite point-sets on 𝕊2 with minimum distance as large as possible,” Discr. Math., 60, 3–66 (1986).
L. Fejes Tóth, “Über die Abschätzung des kürzesten Abstandes zweier Punkte eines auf einer Kugelfläche liegenden Punktsystems,” Jber. Deutsch. Math. Verein., 53, 66–68 (1943).
L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und in Raum, Springer-Verlag, Berlin (1953).
A. V. Kostin, “Shadow problem in the Lobachevsky space,” Ukr. Mat. Zh., 70, No. 11, 1525–1532 (2018).
A. V. Kostin and N. N. Kostina, “An interpretation of Casey’s theorem and of its hyperbolic analog,” Sib. Elektron. Mat. Izv., 13, 242–251 (2016).
V. I. Levenshtein, “On bounds for packing in n-dimensional Euclidean space,” Sov. Math. Dokl., 20, No. 2, 417–421 (1979).
H. Maehara, “Isoperimetric theorem for spherical polygons and the problem of 13 spheres,” Ryukyu Math. J., 14, 1770–1778 (2001).
H. Maehara, “The problem of thirteen spheres - a proof for undergraduates,” Eur. J. Combin., 28, 41–57 (2007).
O. R. Musin, “The kissing problem in three dimensions,” Discr. Comput. Geom., 35, 375–384 (2006).
O. R. Musin, “The kissing problem in four dimensions,” Ann. Math., 168, 1–32 (2008).
O. R. Musin and A. C. Tarasov, “The strong thirteen spheres problem,” Discr. Comput. Geom., 48, 128–141 (2012).
O. R. Musin and A. C. Tarasov, The Tammes problem for N = 14 .
A. M. Odlyzko and N. J. A. Sloane, “New bounds on the number of unit spheres that can touch a unit sphere in n dimensions,” J. Combin. Th. A., 26, 210–214 (1979).
F. Pfender and G. M. Ziegler, “Kissing numbers, sphere packings, and some unexpected proofs,” Not. Am. Math. Soc., 51, 873–883 (2004).
R. M. Robinson, “Arrangement of 24 circles on a sphere,” Math. Ann., 144, 14–48 (1961).
K. Sch¨utte and B. L. van der Waerden, “Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand 1 Platz?,” Math. Ann., 123, 96–124 (1951).
K. Sch¨utte and B. L. van der Waerden, “Das Problem der dreizehn Kugeln,” Math. Ann., 125, 325–334 (1953).
R. M. L. Tammes, “On the origin number and arrangement of the places of exits on the surface of pollen-grains,” Rec. Trv. Bot. Neerl., 27, 1–84 (1930).
V. A. Zinoviev and T. Ericson, “New lower bounds for contact numbers in small dimensions,” Problems Inform. Transmission, 35, No. 4, 287–294 (1999).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 182, Proceedings of the International Conference “Classical and Modern Geometry” Dedicated to the 100th Anniversary of Professor V. T. Bazylev. Moscow, April 22-25, 2019. Part 4, 2020.
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Kostin, A.V., Kostina, N.N. Tammes Problem and Contact Number for Spheres in Spaces of Constant Curvature. J Math Sci 277, 750–755 (2023). https://doi.org/10.1007/s10958-023-06882-4
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DOI: https://doi.org/10.1007/s10958-023-06882-4