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Sails and norm minima of lattices. (English. Russian original) Zbl 1084.11035
Sb. Math. 196, No. 3, 337-365 (2005); translation from Mat. Sb. 196, No. 3, 31-60 (2005).
The main result of this paper is the following Theorem: The norm minimum of an \(n\)-dimensional lattice \(\Lambda \subset \mathbb R^{n}\) is non-zero if and only if there is a uniform bound on the determinants of the faces of each of the \(2^{n}\) sails generated by the lattice \(\Lambda\) and the standard cone \(\mathcal C_{0}: \{(t_{1}, \dots, t_{n}\,| \, t_{i}>0\}\). Here the norm minimum of \(\Lambda\) is defined to be \(\inf_{(x_{1}, \dots, x_{n})\in \Lambda \setminus \{0\}}\; | x_{1}\dots x_{n}| \); a sail is the boundary of the convex hull of the intersection of a cone \(\mathcal C\) with \(\Lambda \setminus \{0\}\); the sails generated by \(\Lambda\) and \(\mathcal C_{0}\) arise from letting \(\mathcal C\) be generated by the various choices for \((\pm e_{1}, \dots, \pm e_{n})\), where the \(e_{i}\) form the canonical basis of \(\mathbb R^{n}\); each sail is a (generalized) \(n-1\) polytope, faces are as usual; the determinant of a face is its normalized volume (so that the determinant of a simplicial face is the determinant of the matrix with entries the components of the vertices).

MSC:
11H50 Minima of forms
11J70 Continued fractions and generalizations
11H06 Lattices and convex bodies (number-theoretic aspects)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
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