Relative volumes and minors in monomial subrings

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