## 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)

### Keywords:

multidimensional continued fractions
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