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.

中文翻译：

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凸体的混合体积之间的不等式：MINKOWSKI 和的体积界限

在对可在根中求解的通用稀疏多项式系统进行分类的过程中，Esterov 最近表明，$d$ 维晶格多胞体的 Minkowski 和 $P_1+\dots+P_d$ 的体积从上方由阶函数 $ O(m^{2^d})$，其中 $m$ 是元组 $(P_1,\dots,P_d)$ 的混合体积。这是众所周知的 Aleksandrov-Fenchel 不等式的结果。Esterov 还提出了确定更锐界的问题。我们展示了如何使用混合体积之间的附加关系来改善渐近尖锐的 $O(m^d)$ 的界限。我们进一步证明了维度 2 和 3 的精确上限。我们的结果推广到体积至少为 1 的任意凸体的元组。