Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T07:11:51.841Z Has data issue: false hasContentIssue false
  • English
  • Français

Radii and the Sausage Conjecture

Published online by Cambridge University Press:  20 November 2018

Károly Bőrőczky Jr.  and
Martin Henk
Károly Bőrőczky Jr.
Affiliation:
MTA Matematikai Kutató Intézet, Budapest, Pf 127, 1364 Hungary, e-mail:h5808bor@ella.hu
Martin Henk
Affiliation:
Mathematisches Institut, Universität Siegen Hölderlinstrasse 3 5900 Siegen Germany e-mail:henk@hrz. uni-siegen.dbp. de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1975, L. Fejes Toth conjectured that in Ed, d ≥ 5, the sausage arrangement is denser than any other packing of n unit balls. This has been known if the convex hull Cn of the centers has low dimension. In this paper, we settle the case when the inner m-radius of Cn is at least O(ln d/m). In addition, we consider the extremal properties of finite ballpackings with respect to various intrinsic volumes.

Keywords

52C1752A40
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 38 , Issue 2 , 01 June 1995 , pp. 156 - 166
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Artin, E., The Gamma function, Holt, Rinehardt and Winston, New York, 1964.Google Scholar
2. Ball, K., Ellipsoids of maximal volume in convex bodies, Geom. Dedicata, 41(1992), 241–250.Google Scholar
3. Betke, U. and Gritzmann, P., Ùber L. Fejes Tôths Wurstvermutung in kleinen Dimensionen, Acta Math. Hungar. 43(1984), 299–307.Google Scholar
4. Betke, U., Gritzmann, P. and Wills, J. M., Slices of L. Fejes Tôths Sausage Conjecture, Mathematika 29 (1982), 194–201.Google Scholar
5. Bôrôzky, K., Jr., Intrinsic volumes of finite ball-packings, Ph.D thesis, University of Calgary, 1992.Google Scholar
6. Conway, J. H. and Sloane, N. J. A., Sphere Packings, Lattices and Groups, Springer-Verlag, Berlin, 1988.Google Scholar
7. Eggleston, H. V., Convexity, Cambridge Univ. Press, Cambridge, 1958.Google Scholar
8. Fejes Tôth, G., Gritzmann, P. and Wills, J. M., Finite sphere packing and sphere covering, Discrete Comput. Geom. 4(1989), 19–40.Google Scholar
9. Fejes Toth, L., Research Problem 13, Period. Math. Hungar. 6(1975), 197–199.Google Scholar
10. Folkman, J. H. and Graham, R. L., A packing inequality for compact, convex subsets of the plane, Canad. Math. Bull. 12(1969), 745–752.Google Scholar
11. Gritzmann, P. and Klee, V., Inner and outer j-radii of convex bodies infinite dimensional normed spaces, Discrete Comput. Geom. 7(1992), 255–280.Google Scholar
12. Gruber, P. M. and Lekkerkerker, C. G., Geometry of Numbers, North-Holland, Amsterdam, 1987.Google Scholar
13. Henk, M., Ungleichunger fur sukzessive Minima and verallgemeinerte In und Umkugelradien, Ph.D thesis, University of Siegen, 1991.Google Scholar
14. Oler, N., An equality in the geometry of numbers, Acta Math. 105(1961), 19–48.Google Scholar
15. Pinkus, A., n-Widths in Approximation Theory, Springer-Verlag, Berlin, 1985.Google Scholar
16. Kleinschmidt, P., Pachner, U. and Wills, J. M., On L. Fejes Tôths ‘Sausage Conjecture', Israel J. Math. 47(1984), 216–226.Google Scholar
17. Pukhov, S. V., Inequalities between the Kolgomorov and the Berstein diameters in a Hilbert space, Math. Notes 25(1979), 320–326.Google Scholar
18. Rankin, R. A., On the closest packing of spheres in n dimensions, Ann. of Math. 48(1947), 1062—1081.Google Scholar