Klein polyhedra and relative minima of lattices
type: Item
instance of: scholarly article
title: Klein polyhedra and relative minima of lattices
zbMATH Open document ID: 1119.11036
published in: Mathematical Notes
publication date: 2007-05-21T00:00:00Z
review text: Let \({\mathcal L}^n\) be the space of \(n\)-dimensional unimodular lattice in \(\mathbb{R}^n\). For a lattice \(\Lambda\in{\mathcal L}^n\), by \(K_\Lambda\) denote the Klein polyhedron, and by \({\mathfrak M}(\Lambda)\) the set of all relative minima of \(\Lambda\). Moreover let \(\text{int}(K_\Lambda)\) be the set of internal points of \(K_\Lambda\), and \(\Lambda^3_{1/2}\) be the minimal lattice (with respect to inclusion) containing \(\mathbb{Z}^3\) and the point \(2^{1/3}(1/2,1/2,1/2)\). Denote by \({\mathcal L}^3_{1/2}\) the orbit of \(\Lambda^3_{1/2}\) under the left action of the group of diagonal unimodular \(3\times 3\) matrices. The author proves that if \(\Lambda\in{\mathcal L}^3\setminus{\mathcal L}^3_{1/2}\), then \({\mathfrak M}(\Lambda)\cap \text{int}(K_\Lambda)= \emptyset\). (Hence the relative minima of almost any lattice belong to the surface of the corresponding Klein polyhedron.) But the similar result is false for the dimension \(n\geq 4\). Moreover, the author also proves that for almost any lattice in \(\mathbb{R}^3\), the set of relative minima with nonnegative coordinates coincides with the union of the set of extremal points of the Klein polyhedron and a set of special points belonging to the triangular faces of the Klein polyhedron.
reviewed by: Yaochen Zhu
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 11H50
Mathematics Subject Classification ID: 52C07
zbMATH DE Number: 5155476
zbMATH Keywords: lattice
zbMATH Keywords: multidimensional continued fraction
zbMATH Keywords: Klein polyhedron
author: Oleg N. German
MaRDI profile type: Publication
full work available at URL: s11006-006-0055-1
OpenAlex ID: W1989943959
cites work: Local minima of lattices and vertices of Klein polyhedra
cites work: Convex hulls of integral points
cites work: Q5341481
cites work: Q4681906
cites work: Lattice Tetrahedra
DOI: 10.1007/S11006-006-0055-1
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description: scientific article; zbMATH DE number 5155476