  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.
