Inequalities between mixed volumes of convex bodies: volume bounds for\n the Minkowski sum

In the course of classifying generic sparse polynomial systems which are\nsolvable in radicals, Esterov recently showed that the volume of the Minkowski\nsum $P_1+\\dots+P_d$ of $d$-dimensional lattice polytopes is bounded from above\nby a function of order $O(m^{2^d})$, where $m$ is the mixed volume of the tuple\n$(P_1,\\dots,P_d)$. This is a consequence of the well-known Aleksandrov-Fenchel\ninequality. Esterov also posed the problem of determining a sharper bound. We\nshow how additional relations between mixed volumes can be employed to improve\nthe bound to $O(m^d)$, which is asymptotically sharp. We furthermore prove a\nsharp exact upper bound in dimensions 2 and 3. Our results generalize to tuples\nof arbitrary convex bodies with volume at least one.\n