Relative volumes and minors in monomial subrings

Let F={xv1,…,xvq} be a finite set of monomials in a polynomial ring R=K[x1,…,xn] over a field K and let P be the convex hull of v1,…,vq. Using linear algebra we show an expression for the relative volume of P. If v1,…,vq lie in a positive hyperplane and the Rees algebra R[Ft] is normal, we prove the equality K[Ft]=A(P), where A(P) is the Ehrhart ring of P and K[Ft] is the monomial subring generated by Ft. We characterize, in terms of minors, when the integral closure of K[Ft] is equal to A(P).