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arXiv · 2002.03065

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

Gennadiy Averkov·Christopher Borger·Ivan Soprunov
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Abstract

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.

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Gennadiy Averkov, Christopher Borger, Ivan Soprunov. 2020-02-08. Inequalities between mixed volumes of convex bodies: volume bounds for the Minkowski sum. https://arxiv.org/abs/2002.03065

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