Polynomial time algorithms to approximate mixed volumes within a simply exponential factor

Let K = (K1...Kn) be a n-tuple of convex compact subsets in the Euclidean space R n, and let V (·) be the Euclidean volume in R n. The Minkowski polynomial VK is defined as VK(x1,..., xn) = V (λ1K1 +... + λnKn) and the mixed volume V (K1,..., Kn) as V (K1...Kn) = ∂n VK(λ1K1 + · · ·λnKn)