## $$k$$-extreme lattices. (Réseaux $$k$$-extrêmes.)(French)Zbl 0861.11040

We study the invariants introduced by R. A. Rankin [J. Lond. Math. Soc. 28, 309-314 (1953; Zbl 0050.27401)] which are defined as a natural generalization of the well-known Hermite constant of a lattice $$L$$. $$k$$ being an integer, one defines $$\delta_k(L)$$ to be the lowest determinant of $$k$$-dimensional subsections in $$L$$, and sets $$\gamma_k (L) = \delta_k (L)/ \text{det} (L)^{k/n}$$.
We give necessary and sufficient conditions for a lattice to realize a local maximum for the function $$\gamma_k(L)$$, analogous to Voronoï’s theorem, in terms of $$k$$-perfection and $$k$$-eutaxy. We also establish the finiteness, up to similarity, of the number of $$k$$-perfect lattices in a given dimension. We finally apply these notions to some classical families of lattices (e.g. root lattices).

### MSC:

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

Zbl 0050.27401
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