Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T19:29:22.333Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on coverings

Published online by Cambridge University Press:  26 February 2010

C. A. Rogers
C. A. Rogers
Affiliation:
The University, Birmingham, 15.
Get access

Extract

Let K be a bounded n-dimensional convex body, with its centroid at the origin o. Let ϑ denote the density of the most economical lattice covering of the whole of space by K (i.e. the lower bound of the asymptotic densities of the coverings of the whole space by a system of bodies congruent and similarly situated to K, their centroids forming the points of a lattice); and let ϑ* denote the density of the most economical covering of the whole space by K (i.e. the lower bound of the asymptotic lower densities of the coverings of the whole space by a system of bodies congruent and similarly situated to K).

Type
Research Article
Information
Mathematika , Volume 4 , Issue 1 , June 1957 , pp. 1 - 6
Copyright
Copyright © University College London 1957

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

page 1 note † For a more precise explanation of these concepts see Hlawka, E., Monatshefte für Math., 53 (1949), 81–131CrossRefGoogle Scholar, or Rogers, C. A., J. London Math. Soc., 25 (1950), 327–331.CrossRefGoogle Scholar A covering is said to have an asymptotic density, if the ratio of the sum of the volumes of the intersections of the bodies of the covering with a large cube to the volume of the cube tends to a limit (called the density of the, covering) as the side of the cube tends to infinity. The asymptotic lower density is defined similarly in terms of a lower limit.

page 1 note ‡ Loc. cit.

page 1 note § J. Indian Math. Soc., 16 (1952), 7–12.Google Scholar

page 1 note || Mat., Rendiconti del C.di Palermo, Series 2, 1 (1952), 92–107.Google Scholar

page 1 note ¶ Mat., Rendiconti del G.di Palermo, Series 2, 5 (1956), 93–100.Google Scholar

page 2 note † J. London Math. Soc., 28 (1953), 287–293;Google Scholar but see also the earlier work of Bambah, R. P. and Davenport, H., J. London Math. Soc., 27 (1952), 224–229CrossRefGoogle Scholar, for the corresponding bound for υ.

page 2 note ‡ We use the usual vector addition notation for the translation of a set by a vector and also for the vector addition of two sets.

page 5 note † For this well known result see Bonnesen, T. and Fenchel, W., Konvexe Körper, Ergebnisse der Mathematik, 3, 1 (Springer, 1934), §34.Google Scholar