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Sur un problème de dualité lié aux sphères en géométrie des nombres. (On a duality problem related to spheres in geometry of numbers). (French) Zbl 0677.10022

Let L be a lattice in \({\mathbb{R}}^ n\) and \(L^*=\{y\in {\mathbb{R}}^ n:\) \(x\cdot y\) integer for all \(x\in L\}\) its dual lattice. Let \(\| L\| =\min \{\| x\|:\) \(x\in L\setminus \{0\}\}\) and \(S(L)=\{x\in L:\) \(\| x\| =\| L\| \}\). Call L dually extreme if among all lattices M sufficiently close to L the expression \(\| M \| \cdot\) \(\| M^*\|\) attains its maximum for \(M=L\). In order to define the concepts of dually perfect and dually eutactic lattices let End be the space of all real symmetric \(n\times n\)-matrices. For \(x\in {\mathbb{R}}^ n\) let \(\phi_ x\) be the linear form on End defined by \(\phi_ x(u)=x^{tr}ux\) for \(u\in End\). The lattice L is dually perfect if the linear forms \(\phi_ x: x\in S(L)\cup S(L^*)\) generate all linear forms on End. It is dually eutactic if there are positive numbers \(\rho_ x: x\in S(L)\), \(\rho_ y: y\in S(L^*)\) such that \(\sum \{\rho_ x\phi_ x: x\in S(L)\}=\sum \{\rho_ y\phi_ y: y\in S(L^*)\}.\)
In the first part of this article the authors give some results on constants introduced by R. A. Rankin [J. Lond. Math. Soc. 28, 309- 314 (1953; Zbl 0050.274)] and related to Hermite’s constant. Then the following analogs of classical theorems due to Korkine and Zolotareff and Voronoi, respectively, are proved: (i) A lattice L is dually extreme if and only if the system of inequalities \(\phi_ x(u)\geq 0: x\in S(L)\), \(\phi_ y(u)\leq 0: y\in S(L^*)\) has only the trivial solution \(u=0\) in End. (ii) A lattice is dually extreme if and only if it is dually perfect and dually eutactic.
In the final part of the paper classical lattices introduced by Korkine and Zolotareff, Coxeter, Barnes and others are investigated. For \(n\leq 4\) all dually extreme lattices are determined. (For additional references consult the reviewer and C. G. Lekkerkerker [Geometry of Numbers (North-Holland (1987; Zbl 0611.10017 and Zbl 0198.380)].)
Reviewer: P.Gruber

MSC:

11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11H06 Lattices and convex bodies (number-theoretic aspects)
11H60 Mean value and transfer theorems
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