Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T07:02:29.594Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-convex regions in three and more dimensions

Published online by Cambridge University Press:  24 October 2008

A. M. Macbeath
A. M. Macbeath
Affiliation:
Clare CollegeCambridge
Get access Rights & Permissions [Opens in a new window]

Extract

Blichfeldt (2) gave a method of proving the existence of lattice points in certain regions. His method depends on the following lemma (which he stated in a somewhat different form):

Lemma 1. Let K′ be a closed region of volume V. Let K be another region so related to K′ that the difference of any two points in K′ is a point in K. Let Λ be any lattice whose determinant is positive and does not exceed V.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 45 , Issue 2 , April 1949 , pp. 161 - 166
Copyright
Copyright © Cambridge Philosophical Society 1949

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

REFERENCES

(1)Artin, E.Einführung in die Theorie der Gammafunktion (Leipzig, 1931).Google Scholar
(2)Blichfeldt, H. F.On a new principle in the geometry of numbers with some applications. Trans. American Math. Soc. 15 (1914), 227–35.CrossRefGoogle Scholar
(3)Davenport, H.On the minimum of x 3 + y 3 + z 3. J. London Math. Soc. 21 (1946), 82–6.CrossRefGoogle Scholar
(4)Koksma, J. F. and Meulenbeld, B.Sur le théorème de Minkowski concernant un système de formes linéaires réelles. Proc. Ned. Acad. Wet., Amsterdam, 5 (1942), 256–62, 354–9, 471–8, 578–84.Google Scholar
(5)Mordell, L. J.Lattice points in some n-dimensional non-convex regions. Proc. Ned. Acad. Wet., Amsterdam, 7 (1946), 773–81, 782–92.Google Scholar
(6)Bonnesen, T. and Fenchel, W.Theorie der konvexen Körper (Ergebnisse der Math.) (Berlin, 1934).Google Scholar