Abstract
We show that for any acute ϕ, there exists a covering of S d by spherical balls of radius ϕ such that no point is covered more than 400d ln d times. It follows that the density is of order at most d ln d, and even at most d ln ln d if the number of balls is polynomial in d. If the number of equal spherical balls is d + 3 then we determine the optimal arrangement.
At the end, we described how our and other peoples results yield estimates for the largest origin centred Euclidean ball contained in the convex hull of N points chosen from the sphere.
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References
E. Artin: The gamma function. Holt, Rinehart and Winston, 1964.
I. B´ar´any: The densest (n+2) set in Rn. In: Intuitive Geometry (1991), Coll.Math. Soc. J´anos Bolyai, No 63, North Holland, 1997.
I. B´ar´any, Z. F¨uredi: Approximation of the sphere by polytopes having few vertices. Proc. Amer. Math. Soc.,102 (1988), 651–659.
U. Betke, P. Gritzmann, J. M. Wills: Slices of L. Fejes T´oth’s sausage conjecture.Mathematika, 29 (1982), 194–201
T. Bonnesen, W. Fenchel. Theory of Convex bodies. BCS Assoc., Moscow (Idaho), 1987. Translated from German: Theorie der konvexen K¨orper, Springer, 1934.
B. Carl: Inequalities of Bernstein–Jackson–type and the degree of compactness of operators in Banach spaces. Ann. Inst. Fourier (Grenoble), 35 (1985), 79–118.
B. Carl, A. Pajor: Gelfand numbers of operators with values in Hilbert space.Invent. Math., 94 (1988), 479–504.
J.H. Conway, N.J.A. Sloane: Sphere packings, Lattices and Groups. Springer_Verlag, 1989.
H.S.M. Coxeter, L. Few, C.A. Rogers: Covering space with equal spheres.Mathematika, 6 (1959), 147–157.
L. Dalla, D.G. Larman, P. Mani–Levitska, Ch. Zong: The blocking numbers of convex bodies. Discrete Comput. Geom., 24 (2000), 267–277.
Gy. Elekes: Ageom etric inequality and the complexity of computing the volume. Disc. Comp. Geom., 1 (1996), 289–292.
P. Erdőos, C.A. Rogers: Covering space with convex bodies. Acta Arith., 7(1962), 281–285.
G. Fejes T´oth and W. Kuperberg: Packing and covering. In: Handbook of Convex Geometry, P.M. Gruber and J.M. Wills (eds), North Holland, 1993,799–860.
L. Fejes T´oth: Regular Figures. Pergamon Press, 1964.
P.M. Gruber, C.G. Lekkerkerker: Geometry of Numbers. North Holland, 1987.
M. Kochol: Constructive approximation of a ball by polytopes. Math. Slovaca,44 (1994), 99–105.
C.A. Rogers. A note on coverings. Mathematika, 4 (1957), 1–6.
] C.A. Rogers: Covering a sphere with spheres. Mathematika, 10 (1963), 157–164.
[19] K. Sch¨utte: ¨Uberdeckungen der Kugel mit h¨ochstens acht Kreisen. Math.Ann., 129 (1955), 181–186.
Ch. Zong: Sphere packings. Springer–Verlag, 1999.
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Böröczky, K., Wintsche, G. (2003). Covering the Sphere by Equal Spherical Balls. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds) Discrete and Computational Geometry. Algorithms and Combinatorics, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55566-4_10
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DOI: https://doi.org/10.1007/978-3-642-55566-4_10
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