  Minkowski's second theorem can be stated as an inequality for $n$-dimensional
flat Finsler tori relating the volume and the minimal product of the lengths of
closed geodesics which form a homology basis. In this paper we show how this
fundamental result can be promoted to a principle holding for a larger class of
Finsler manifolds. This includes manifolds for which first Betti number and
dimension do no necessarily coincide, a prime example being the case of
surfaces. This class of manifolds is described by a non-vanishing condition for
the hyperdeterminant reduced modulo $2$ of the multilinear map induced by the
fundamental class of the manifold on its first ${\mathbb Z}_2$-cohomology group
using the cup product.
