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Dual systems of integral vectors. (General questions, applications to the geometry of positive quadratic forms). (Russian) Zbl 0751.52007

This paper is in connection with the theory of \(L\)-decompositions of the \(n\)-dimensional lattice-like point systems. The authors introduce some new concepts for the set of integral vectors, they define the “dual” and “closed” systems and the “arithmetic \(L\)-body”. Using the basic properties of these concepts they examine the following question, whether a given arithmetic \(L\)-body can be realized in the \(n\)-dimensional Euclidean space or not. For small \(n\) \((n\leq 5)\) the authors have solved this problem and described all affine types of \(L\)-polyhedra characterizing the \(n\)-dimensional lattices.

MSC:

52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
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