Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-19T20:24:21.524Z Has data issue: false hasContentIssue false
  • English
  • Français

A density bound for efficient packings of 3-space with centrally symmetric convex bodies

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Edwin H. Smith
Edwin H. Smith
Affiliation:
Department MCIS, Jacksonville State University, Jacksonville, AL 36265, U.S.A.
Get access

Abstract

It is shown that every compact convex set K which is centrally symmetric and has a non-empty interior admits a packing of Euclidean 3-space with density at least 0.46421 … The best such bound previously known is 0.30051 … due to the theorem of Minkowski-Hlawka. It is probable that there is such a lower bound which is significantly greater than the one shown in this note, since there is a packing of congruent spheres which has density

MSC classification

Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Mathematika , Volume 46 , Issue 1 , June 1999 , pp. 137 - 144
Copyright
Copyright © University College London 1999

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.Doheny, K.. A higher lower bound for packing density of convex bodies in the plane. Beiträge Alg. Geom., 36 (1995), 109–117.Google Scholar
2.Dowker, C. H.. On minimum circumscribed polygons. Bull. Amer. Math. Soc, 50 (1944), 120–122.CrossRefGoogle Scholar
3.Ennola, V.. On the lattice constant of a symmetric convex domain, J. London Math. Soc, 36 (1961), 135–138.CrossRefGoogle Scholar
4.Fejes Toth, L.. Some packing and covering theorems. Ada Sci. Math. Szeged, 12/A (1950), 62–67.Google Scholar
5.Toth, L. Fejes. Lagerungen in der Ebene, auf der Kugel und im Raum. (Springer, Berlin, 1972).CrossRefGoogle Scholar
6.Hlawka, E.. Zur Geometrie der Zahlen. Math. Z., 49 (1944), 285–312.CrossRefGoogle Scholar
7.Kuperberg, G. and Kuperberg, W.. Double-lattice packings of convex bodies in the plane, Discrete Comput. Geom., 5 (1990), 389–397.CrossRefGoogle Scholar
8.Reinhardt, K.. Uber die dichteste gitterformige Lagerung kongruenter Bereiche in der Ebene und eine besondere Art konvexer Kurven. Abh. math. Sem. hansischer Univ., 10 (1934), 216–230.CrossRefGoogle Scholar
9.Tammela, P.. An estimate of the critical determinant of a two-dimensional convex symmetric domain (Russian), Izv. Vyss. Ucebn. Zaved. Mat., 12 (1970), 103–107.Google Scholar