There is a theorem of Minkowski that says that given $k$ unit vectors $x_i$ that span $\mathbb{R}^n$ and $k$ positive real numbers $a_i$ such that $\sum_{i=0}^k a_i x_i = 0$ then there exists a unique convex polytope (up to translation) such that the $i$th face is normal to $x_i$ and has area ($n-1$ dimensional volume) $a_i$. Does there exist an algorithm constructing this polytope?
In theory it should be straightforward as the area of each face is a piecewise polynomial function of the positions of the half planes describing the other faces. Unfortunately these functions appear to be somewhat problematic to derive, let alone simultaneously solve.