Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T09:51:54.585Z Has data issue: false hasContentIssue false

On Convex Fundamental Regions for a Lattice

Published online by Cambridge University Press:  20 November 2018

A. M. Macbeath*
Affiliation:
Queen's College, Dundee
Rights & Permissions [Opens in a new window]

Extract

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.

Let ∧ be a lattice in Euclidean n-space, that is, ∧ is a set of points ε1a1 + … + εnan where a1… , an are linearly independent vectors and the ε run over all integers. Let μ denote the Lebesgue measure. A closed convex set F is called a fundamental region for ∧ if the sets F + x (x ∈∧) cover the whole space without overlapping; that is, if F0 is the interior of F, and 0 ≠ x ∈ ∧, then F0(F0 + x) = ϕ.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Bonnesen, T. and Fenchel, W., Théorie der konvexen Kôrper (Berlin: Springer, 1934).Google Scholar
2. Macbeath, A. M., Abstract theory of packings and coverings, Proc. Glasgow Math. Assoc, 4 (1959), 9295.Google Scholar
3. Minkowski, H., Allgemeine Lehrsàtze ueber die konvexen Polyeder, Ges. Math. Abh., 2 (1911), 103121.Google Scholar
4. Voronoi, G., Nouvelles applications des paramètres continus à la théorie des formes quadratiques, IL J. reine angew. Math., 134 (1908), 198287.Google Scholar