  We compare for an $n$-dimensional Euclidean lattice $\Lb$ the smallest
possible values of the product of the norms of $n$~vectors which either
constitute a basis for $\Lb$ (Hermite-type inequalities) or are merely assumed
to be independent (Minkowski-type inequalities). We improve on 1953 results of
van der Waerden in dimensions $6$ to $8$ and prove partial result in
dimension~$9$.
