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In the course of classifying generic sparse polynomial systems which aresolvable in radicals, Esterov recently showed that the volume of the Minkowskisum $P_1+\dots+P_d$ of $d$-dimensional lattice polytopes is bounded from aboveby a function of order $O(m^{2^d})$, where $m$ is the mixed volume of the tuple$(P_1,\dots,P_d)$. This is a consequence of the well-known Aleksandrov-Fenchelinequality. Esterov also posed the problem of determining a sharper bound. Weshow how additional relations between mixed volumes can be employed to improvethe bound to $O(m^d)$, which is asymptotically sharp. We furthermore prove asharp exact upper bound in dimensions 2 and 3. Our results generalize to tuplesof arbitrary convex bodies with volume at least one.

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