  In this paper we deduce a lower bound for the rank of a family of $p$ vectors
in $\R^k$ (considered as a vector space over the rationals) from the existence
of a sequence of linear forms on $\R^p$, with integer coefficients, which are
small at $k$ points. This is a generalization to vectors of Nesterenko's linear
independence criterion (which corresponds to $k=1$), used by Ball-Rivoal to
prove that infinitely many values of Riemann zeta function at odd integers are
irrational. The proof is based on geometry of numbers, namely Minkowski's
theorem on convex bodies.
