Hermite versus Minkowski

We compare for an $n$-dimensional Euclidean lattice $\text{Lb}$ the smallest possible values of the product of the norms of $n$~vectors which either constitute a basis for $\text{Lb}$ (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~$9$.