Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-23T05:18:51.568Z Has data issue: false hasContentIssue false
  • English
  • Français

On the sausage catastrophe in 4-space

Part of: Discrete geometry

Published online by Cambridge University Press:  26 February 2010

Pier Mario Gandini  and
Andreana Zucco
Pier Mario Gandini
Affiliation:
Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10, 10123 Torino, Italy.
Andreana Zucco
Affiliation:
Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy.
Get access

Extract

An upper bound for the “sausage catastrophe” of dense sphere packings in 4-space is given.

A basic problem in the theory of finite packing is to determine, for a given positive integer k, the minimal volume of all convex bodies into which k translates of the unit ball Bd of the Euclidean d-dimensional space Ed can be packed ([5]). For d = 2 this problem was solved by Groemer ([6]).

MSC classification

Secondary: 52C17: Packing and covering in $n$ dimensions
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 274 - 278
Copyright
Copyright © University College London 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Betke, U.Gritzmann, P. and Wills, J. M.. Slices of L. Fejes Tóth's sausage conjecture. Mathematika, 29 (1982), 194–201.CrossRefGoogle Scholar
2.Coxeter, H. S. M.. Regular Polytopes. 2nd Edition (MacMillan, New York-London, 1963).Google Scholar
3.Tóth, L. Fejes. Research Problem 13. Period. Math. Hungar., 6 (1975), 197–199.Google Scholar
4.Gritzmann, P.. Finite packing of equal balls. J London Math. Soc., 33 (1986), 543–553.CrossRefGoogle Scholar
5.Gritzmann, P. and Wills, J. M.. Finite Packing and Covering. Studia Sci. Math. Hungarica, 21 (1986), 149–162.Google Scholar
6.Groemer, H.. Über die Einlagerung von Kreisen in einen konvexen Bereich. Math. Zeitschrift, 73 (1960), 285–294.CrossRefGoogle Scholar
7.McMullen, P. and Schneider, R.. Valuations on convex bodies. Convexity and its Applications, Edited by Gruber, P. and Wills, J. M. (Birkhauser, 1983), 170–247.CrossRefGoogle Scholar
8.Rogers, C. A.. Packing and Covering (Cambridge Univ. Press, 1964).Google Scholar
9.Wills, J. M.. On the density of finite packing. Acta Math. Hung., 46 (1985), 205–210.CrossRefGoogle Scholar