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Klein polyhedra and lattices with positive norm minima. (English) Zbl 1180.11023
Summary: A Klein polyhedron is defined as the convex hull of nonzero lattice points inside an orthant of $$\mathbb{R}^n$$. It generalizes the concept of continued fraction. In this paper facets and edge stars of vertices of a Klein polyhedron are considered as multidimensional analogs of partial quotients and quantitative characteristics of these “partial quotients”, so called determinants, are defined. It is proved that the facets of all the $$2^n$$ Klein polyhedra generated by a lattice $$\Lambda$$ have uniformly bounded determinants if and only if the facets and the edge stars of the vertices of the Klein polyhedron generated by $$\Lambda$$ and related to the positive orthant have uniformly bounded determinants.

##### MSC:
 11H06 Lattices and convex bodies (number-theoretic aspects) 52C07 Lattices and convex bodies in $$n$$ dimensions (aspects of discrete geometry) 11A55 Continued fractions
Klein polyhedra
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##### References:
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