A Survey on Voronoï’s Theorem
نویسنده
چکیده
Let Vn be the vector space of real n×n symmetric matrices and Pn the open cone of positive definite symmetric matrices in Vn. By m1(a), we denote the arithmetical minimum infx∈Zn\{0} xax of a ∈ Pn. The Hermite invariant is the positive valued function γ on Pn defined by γ(a) = m1(a)/det(a). Its maximum γn is called Hermite’s constant. The determination of γn is one of main problems in lattice sphere packings or the arithmetic theory of quadratic forms. Voronöı’s fundamental theorem [61] gives a charactrization of local maxima of γ, i.e., which can be stated that γ attains a local maximum on a ∈ Pn if and only if a is perfect and eutactic. In the last half of 20th century, various generalizations of Hermite’s constant and Voronöı’s theorem were studied by many authors. In this paper, we give an account of a recent development concerning Voronöı’s theorem.
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