On linear sections of lattice packings
type: Item
instance of: scholarly article
title: On linear sections of lattice packings
zbMATH Open document ID: 0598.52009
DOI: 10.1007/BF01294599
published in: Monatshefte für Mathematik
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178299
review text: In d-dimensional Euclidean space \(E^ d\) let \({\mathfrak P}\) be a lattice packing of subsets of \(E^ d\), and let H be a k-dimensional linear subspace of \(E^ d\) \((0<k<d)\). Then, \({\mathfrak P}\) induces a packing in H consisting of all sets \(P\cap H\) with \(P\in {\mathfrak P}\). The relationship between the density of this packing in H and the density of \({\mathfrak P}\) is investigated. A result from the theory of uniform distribution of linear forms is used to prove an integral formula that enables one to valuate the density of the induced packing in H (under suitable assumptions on the sets of \({\mathfrak P}\) and the functionals used to define the densities). It is shown that this result leads to explicit formulas for the averages of the induced densities under the rotation of H. If the densities are taken with respect to the mean cross-sectional measures of convex bodies one obtains analogues of the integral geometric intersection formulas of Crofton.
Mathematics Subject Classification ID: 52C07
Mathematics Subject Classification ID: 11H31
Mathematics Subject Classification ID: 11K06
zbMATH DE Number: 3964872
zbMATH Keywords: convex set
zbMATH Keywords: lattice packing of subsets
zbMATH Keywords: density
zbMATH Keywords: uniform distribution
author: Helmut Groemer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3964872