The authors study the Newton polyhedra of the polynomial given by the product of all maximal minors of a \(m \times n\) matrix of indeterminates \(X=(x_{ij})\). It is a polytope in \(\mathbb{R}^{mn}\). The description of this polytope is well known in the following cases:
If \(m=n\) it is the Birkhoff polytope of doubly stochastic \(n \times n\) matrices. – If \(m=2\) it is the convex hull in \(\mathbb{R}^{2n}\) of all \(n!\) matrices obtained from \(\begin{pmatrix} n-1 & n-2 & \ldots & 1 & 0 \\ 0& 1 & \ldots & n-2 & n-1 \end{pmatrix}\) by permuting columns. The description of this polytope is really difficult and interesting. The authors give some motivations and applications.