A Polynomial-Time Algorithm to Approximate the Mixed Volume within a Simply Exponential Factor

Let K=(K1,…,Kn) be an n-tuple of convex compact subsets in the Euclidean space Rn, and let V(⋅) be the Euclidean volume in Rn. The Minkowski polynomial VK is defined as VK(λ1,…,λn)=V(λ1K1+⋅⋅⋅+λnKn) and the mixed volume V(K1,…,Kn) as $$V(K_{1},\ldots,K_{n})=\frac{\partial^{n}}{\partial\lambda_{1}\cdots\partial \lambda_{n}}V_{\mathbf{K}}(\lambda_{1},\ldots,\lambda_{n}).$$ Our main result is a poly-time algorithm which approximates V(K1,…,Kn) with multiplicative error en and with better rates if the affine dimensions of most of the sets Ki are small. Our approach is based on a particular approximation of log (V(K1,…,Kn)) by a solution of some convex minimization problem. We prove the mixed volume analogues of the Van der Waerden and Schrijver–Valiant conjectures on the permanent. These results, interesting on their own, allow us to justify the abovementioned approximation by a convex minimization, which is solved using the ellipsoid method and a randomized poly-time time algorithm for the approximation of the volume of a convex set.