We prove the theorem mentioned in the title, for \({\mathbb{R}}^n\), where \(n \ge 3\). The case of the simplex was known previously. Also, the case \(n=2\) was settled, but there the infimum was some well-defined function of the side lengths. We also consider the cases of spherical and hyperbolic \(n\)-spaces. There we give some necessary conditions for the existence of a convex polytope with given facet areas, and some partial results about sufficient conditions for the existence of (convex) tetrahedra.