  In this paper, motivated by the work of Edelman and Strang, we show that for
fixed integers $d\geq 2$ and $n\geq d+1$ the configuration space of all facet
volume vectors of all $d$-polytopes in $\mathbb R^{d}$ with $n$ facets is a
full dimensional cone in $\mathbb R^{n}$. In particular, for tetrahedra ($d=3$
and $n=4$) this is a cone over a regular octahedron. Our proof is based on a
novel configuration space / test map scheme which uses topological methods for
finding solutions of a problem, and tools of differential geometry to identify
solutions with the desired properties. Furthermore, our results open a
possibility for the study of realization spaces of all $d$-polytopes in
$\mathbb R^{d}$ with $n$ facets by the methods of algebraic topology.
