Isoperimetric inequalities for densities of lattice-periodic sets
type: Item
instance of: scholarly article
title: Isoperimetric inequalities for densities of lattice-periodic sets
zbMATH Open document ID: 0931.52006
published in: Monatshefte für Mathematik
publication date: 1999-06-10T00:00:00Z
full work available at URL: https://refubium.fu-berlin.de/handle/fub188/19095
review text: Lattice-periodic sets have infinite volume and surface area. These can be replaced by appropriate finite measures, the volume density and the surface area density, both measured with respect to the fundamental cell. For these densities there exist isoperimetric inequalities, analogously to the classical ones. The first one was discovered by H. Hadwiger in 1972 for the standard lattice \(\mathbb{Z}^d\), and he determined also the extremal cases. In 1993 U. Schnell found the corresponding isoperimetric inequality for general lattices in terms of Minkowski's last successive minimum. Here the author gives another interesting isoperimetric inequality in terms of minimal determinants of sublattices; proved for \(d=2\) and conjectures for \(d\geq 3\).
reviewed by: Jörg M. Wills
Mathematics Subject Classification ID: 52C07
Mathematics Subject Classification ID: 52C05
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 1300154
zbMATH Keywords: densities of lattice-periodic sets
zbMATH Keywords: isoperimetric inequalities
zbMATH Keywords: minimal determinants of sublattices
author: Peter Brass
MaRDI profile type: Publication
OpenAlex ID: W2116680421
DOI: 10.1007/S006050050032
description: scientific article; zbMATH DE number 1300154