Abstract
A basic problem of finite packing and covering is to determine, for a given number ofk unit balls in Euclideand-spaceE d, (1) the minimal volume of all convex bodies into which thek balls can be packed and (2) the maximal volume of all convex bodies which can be covered by thek balls. In the sausage conjectures by L. Fejes Tóth and J. M. Wills it is conjectured that, for alld≥5, linear arrangements of thek balls are best possible. In the paper several partial results are given to support both conjectures. Furthermore, some relations between finite and infinite (space) packing and covering are investigated.
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References
U. Betke and P. Gritzmann, Über L. Fejes Tóths Wurstvermutung in kleinen Dimensionen,Acta Math. Hungar. 43 (1984), 299–307.
U. Betke, P. Gritzmann, and J. M. Wills, Slices of L. Fejes Tóth's sausage conjecture,Mathematika 29 (1982), 194–201.
H. F. Blichfeldt, The minimum value of quadratic forms and the closest packing of spheres,Math. Ann. 101 (1929), 605–608.
H. S. M. Coxeter, L. Few, and C. A. Rogers, Covering space with equal spheres,Mathematika 6 (1959), 147–157.
L. Fejes Tóth, Über die dichteste Kreislagerung und dünnste Kreisüberdeckung,Comment. Math. Helv. 23 (1949), 342–349.
L. Fejes Tóth,Lagerungen in der Ebene, auf der Kugel und im Raum, Springer-Verlag, Berlin, 1953, 1972.
L. Fejes Tóth, Research problem 13,Period. Math. Hungar. 6 (1975), 197–199.
G. Fejes Tóth, New results in the theory of packing and covering, inConvexity and Its Applications (P. M. Gruber and J. M. Wills, eds.), 318–359, Birkhäuser, Basel, 1983.
P. Gritzmann and J. M. Wills, On two finite covering problems of Bambah, Rogers, Woods, and Zassenhaus,Monatsh. Math. 99 (1985), 279–296.
P. Gritzmann and J. M. Wills, Finite packing and covering,Studia Sci. Math. Hungar. 21 (1986), 151–164.
H. Groemer, Über die Einlagerung von Kreisen in einen konvexen Bereich,Math. Z. 73 (1960), 285–294.
H. Hadwiger,Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin, 1957.
H. Hadwiger, Gitterpunktanzahl im Simplex und Wills'sche Vermutung,Math. Ann. 239 (1979), 271–288.
G. A. Kabatjanski and V. I. Levenstein, Bounds for packings on the sphere and in space,Problemy Peredachi Informaticii 14 (1978), 3–25 (Russian). (English transl.;Problems Inform. Transmission 14 (1978), 1–17.)
J. Leech, Some sphere packings in higher space,Canad. J. Math. 16 (1964), 657–682.
P. McMullen, Non-linear angle-sum relations for polyhedral cones and polytopes,Math. Proc. Cambridge Philos. Soc. 78 (1975), 247–261.
A. M. Odlyzko and N. J. A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere inn dimensions,J. Combin. Theory Ser. A 26 (1979), 210–214.
R. A. Rankin, The closest packing of spherical caps inn dimensions,Proc. Glasgow Math. Assoc. 2 (1955), 139–144.
C. A. Rogers, The packing of equal spheres,Proc. London Math. Soc. (3),8 (1958), 609–620.
C. A. Rogers,Packing and Covering, Cambridge University Press, Cambridge, 1964.
H. Ruben, On the geometrical moments of skew-regular simplices in hyperspherical space; with some applications in geometry and mathematical statistics,Acta Math. 103 (1960), 1–23.
G. Wegner, Über endliche Kreispackungen in der Ebene,Studia Sci. Math. Hungar., in press.
J. M. Wills, Research problems 30, 33, and 35,Period. Math. Hungar. 13 (1982), 75–76,14 (1983), 189–191, 312–314.
J. M. Wills, On the density of finite packings,Acta Math. Hungar. 46 (1985), 205–210.
J. M. Wills, Research problem 41: space conjecture for finite packing and covering,Period. Math. Hungar. 18 (1987), 251–252.
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This paper was written while the first named author was visiting the “Forschungsinstitut für Geistes- und Sozialwissenschaften” at the University of Siegen.
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Fejes Tóth, G., Gritzmann, P. & Wills, J.M. Finite sphere packing and sphere covering. Discrete Comput Geom 4, 19–40 (1989). https://doi.org/10.1007/BF02187713
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DOI: https://doi.org/10.1007/BF02187713