Volume of Minkowski Sums with Fixed Mixed Volume

Definitions and mathematical statement

Let $d\geq 1$ be an integer. A full-dimensional lattice polytope in ${\mathbb{R}}^d$ is the convex hull of finitely many points of ${\mathbb{Z}}^d$ whose affine span is ${\mathbb{R}}^d$. Write $\operatorname{vol}_d$ for Euclidean volume, with the standard lattice fundamental parallelepiped having volume $1$, and put \[ \operatorname{Vol}_{{\mathbb{Z}}}(P)=d!\operatorname{vol}_d(P). \] For polytopes $P_1,\ldots,P_d$, their Minkowski sum is \[ P_1+\cdots+P_d=\{x_1+\cdots+x_d:x_i\in P_i\}. \] Use the standard normalized mixed volume convention \[ V(P_1,\ldots,P_d) =[t_1\cdots t_d]\,\operatorname{vol}_d(t_1P_1+\cdots+t_dP_d), \] where the brackets denote coefficient extraction from the volume polynomial. Thus $V(P,\ldots,P)=\operatorname{Vol}_{{\mathbb{Z}}}(P)$; for full-dimensional lattice polytopes the mixed volume is a positive integer. The supplied record does not explicitly specify its normalization, so this convention requires confirmation against the cited source.

The question is whether, for every integer $d\geq1$ and every integer $m\geq1$, \[ \sup\left\{ \operatorname{Vol}_{{\mathbb{Z}}}(P_1+\cdots+P_d): \begin{array}{l} P_1,\ldots,P_d\subset{\mathbb{R}}^d\text{ are full-dimensional lattice polytopes},\\ V(P_1,\ldots,P_d)=m \end{array} \right\} =(m+d-1)^d. \] Equivalently, is the displayed quantity an upper bound for every such tuple and the least possible upper bound for each fixed pair $(d,m)$? This is a statement-only formulation; no mathematical verification or solution is asserted.

Short English statement

For each dimension and fixed normalized mixed volume, ask whether the largest possible normalized volume of the Minkowski sum has sharp upper bound $(m+d-1)^d$. Each constituent polytope must have full dimension. The supplied dated assessment labels the problem partially solved; that assessment is not independently verified here.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30003608$, title "Volume of Minkowski Sums with Fixed Mixed Volume," source code catalogue entry. The supplied record gives proposed year $2017$ and names no proposer.
• Supplied report citation: Mini-Workshop: Lattice Polytopes: Methods, Advances, Applications (2018), Oberwolfach Reports, pp. 2659--2701, DOI 10.4171/owr/2017/44. https://doi.org/10.4171/owr/2017/44.
• Supplied literature reference: "Averkov Borger Soprunov 2020." https://doi.org/10.1112/mtk.12055.
• Supplied literature reference: "Averkov Borger Soprunov 2021." https://doi.org/10.1007/s00454-020-00246-4.
• Literature assessment checked on 2026-08-21: the supplied curation states that the conjectured sharp upper bound is proved in dimensions two and three, that general bounds of the correct order are known, and that the all-dimensional sharp equality remains unresolved. These are supplied dated curation claims, not independently verified here; the record expressly calls for independent verification of time-sensitive status and cited claims.