Géométrie des nombres adélique et lemmes de Siegel généralisés

A Siegel’s lemma provides an explicit upper bound for a non-zero vector of minimal height in a finite dimensional vector spaces over a number field. This article explains how to obtain Siegel’s lemmas for which the minimal vectors do not belong to a finite union of vector subspaces (Siegel’s lemmas with conditions). The proofs mix classical results of adelic geometry of numbers and an adelic variant of a theorem of Henk about the number of lattice points of a centrally symmetric convex body in terms of the successive minima of the body.