Volume of Alcoved Polyhedra and Mahler Conjecture

The facet equations of a 3--dimensional alcoved polyhedron P are only of two types (xi=cnst and xi-xj=cnst) and the f --vector of P is bounded above by (20,30,12). In general, P is a dodecahedron with 20 vertices and 30 edges. We represent an alcoved polyhedron by a real square matrix A of order 4 and we compute the exact volume of P: it is a polynomial expression in the aij, homogeneous of degree 3 with rational coefficients. Then we compute the volume of the polar P o, when P is centrally symmetric. Last, we show that Mahler conjecture holds in this case: the product of the volumes of P and Po is no less that 43/3!, with equality only for boxes. Our proof reduces to computing a certificate of non--negativeness of a certain polynomial (in 3 variables, of degree 6, non homogeneous) on a certain simplex.