Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-23T04:53:40.267Z Has data issue: false hasContentIssue false
  • English
  • Français

On successive minima and intrinsic volumes

Part of: Discrete geometry

Published online by Cambridge University Press:  26 February 2010

U. Schnell  and
J. M. Wills
U. Schnell
Affiliation:
Mathematisches Institut, Universität Siegen, 57068 Siegen, Germany.
J. M. Wills
Affiliation:
Mathematisches Institut, Universität Siegen, 57068 Siegen, Germany.
Get access

Abstract

In Euclidean d-space Ed we prove inequalities between the intrinsic volumes (i.e., normalized quermassintegrals) of convex bodies and the successive minima of arbitrary lattices. The inequalities are tight and they generalize earlier results of Hadwiger and Henk for the integer lattice ℤd.

MSC classification

Secondary: 52C07: Lattices and convex bodies in $n$ dimensions
Type
Research Article
Information
Mathematika , Volume 40 , Issue 1 , June 1993 , pp. 144 - 147
Copyright
Copyright © University College London 1993

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.Bokowski, J., Hadwiger, H. and Wills, J. M.. Eine Ungleichung zwischen Volumen, Oberfläche und Gitterpunktanzahl konvexer Körper im n-dimensionalen Raum. Math. Z., 127 (1972), 363–364.Google Scholar
2.GruberandC, P. M.. Lekkerkerker, G.. Geometry of Numbers (North Holland, Amsterdam, 1987).Google Scholar
3.Henk, M.. Inequalities between successive minima and intrinsic volumes of a convex body. Monatsh. Math., 110 (1990), 279–282.Google Scholar
4.Lagarias, J. C., Lenstra, H. W. Jr. and Schnorr, C. P.. Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica, 10 (1990), 333–348.Google Scholar
5.McMullen, P.. Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc. Comb. Phil. Soc., 78 (1975), 247–261.Google Scholar
6.McMullen, P.. Inequalities between Intrinsic Volumes. Monatsh. Math., 111 (1991), 47–53.Google Scholar
7.Schnell, U.. Minimal determinants and lattice inequalities. Bull London Math. Soc., 24 (1992), 606–612.Google Scholar
8.Schnell, U. and Wills, J. M.. Two isoperimetric inequalities with lattice constraints. Monatsh. Math., 112 (1991), 227–233.Google Scholar