  In the course of classifying generic sparse polynomial systems which are
solvable in radicals, Esterov recently showed that the volume of the Minkowski
sum $P_1+\dots+P_d$ of $d$-dimensional lattice polytopes is bounded from above
by 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-Fenchel
inequality. Esterov also posed the problem of determining a sharper bound. We
show how additional relations between mixed volumes can be employed to improve
the bound to $O(m^d)$, which is asymptotically sharp. We furthermore prove a
sharp exact upper bound in dimensions 2 and 3. Our results generalize to tuples
of arbitrary convex bodies with volume at least one.
