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A note on coverings

Published online by Cambridge University Press:  26 February 2010

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

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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