Abstract:
Let ${\bf K} = (K_1...K_n)$ be a $n$-tuple of convex compact subsets in the Euclidean space $\R^n$, and let $V(\cdot)$ be the Euclidean volume in $\R^n$. The Minkowski polynomial $V_{{\bf K}}$ is defined as $V_{{\bf K}}(x_1,...,x_n) = V(\lambda_1 K_1 + ... + \lambda_n K_n)$ and the mixed volume $V(K_1,...,K_n)$ as $$ V(K_1...K_n) = \frac{\partial^n}{\partial \lambda_1...\partial \lambda_n} V_{{\bf K}}(\lambda_1 K_1 + \cdots \lambda_n K_n). $$ In this paper, we study randomized algorithms to approximate the mixed volume of well-presented convex compact sets. Our main result is a polynomial time algorithm which approximates $V(K_1,...,K_n)$ with a multiplicative error of $e^n$ and with better rates if the affine dimensions of most of the sets $K_i$ are small. Our approach is based on a particular convex relaxation of $\log(V(K_1,...,K_n))$ via geometric programming. We prove the mixed volume analogues of the Van der Waerden and the Schrijver/Valiant conjectures for the permanent. These results, though interesting on their own, allow one to "justify" the above mentioned convex relaxation. This relaxation is solved with the ellipsoid method using a randomized polynomial time algorithm for the approximation of the volume of a convex set.

Abstract:
We study in this paper randomized algorithms to approximate the mixed volume of well-presented convex compact sets. Our main result is a poly-time algorithm which approximates $V(K_1,...,K_n)$ with multiplicative error $e^n$ and with better rates if the affine dimensions of most of the sets $K_i$ are small.\\ Our approach is based on the particular convex relaxation of $\log(V(K_1,...,K_n))$ via the geometric programming. We prove the mixed volume analogues of the Van der Waerden and the Schrijver/Valiant conjectures on the permanent. These results , interesting on their own, allow to "justify" the above mentioned convex relaxation, which is solved using the ellipsoid method and a randomized poly-time time algorithm for the approximation of the volume of a convex set.