Successive minima

Summary

The successive minima of a lattice with respect to a symmetric convex body are $n$ numbers $\lambda_1 \le \dots \le \lambda_n$. The $k$-th one is the smallest scaling factor at which the scaled body contains $k$ linearly independent lattice vectors. They measure how long the short vectors of the lattice are, in all independent directions at once. Minkowski's second theorem bounds their product in terms of the volume of the body and the covolume of the lattice.