Unimodality and Log-Concavity under Integer Decomposition

Definitions and mathematical statement

Let $L$ be a lattice of rank $d\geq 0$, let $L_{{\mathbb{R}}}=L\otimes_{{\mathbb{Z}}}{\mathbb{R}}$, and let $P\subseteq L_{{\mathbb{R}}}$ be a nonempty, full-dimensional lattice polytope, meaning the convex hull of finitely many points of $L$. A polytope of smaller dimension in an ambient lattice is understood in its affine span, translated by a lattice vertex and equipped with the induced lattice.

The polytope $P$ has the integer-decomposition property (IDP) if, for every integer $m\geq 1$ and every $x\in mP\cap L$, there exist $x_1,\ldots,x_m\in P\cap L$ such that \[ x=x_1+\cdots+x_m. \] Its Ehrhart $h^*$-polynomial is defined by \[ \sum_{m=0}^{\infty}\lvert mP\cap L\rvert z^m =\frac{h^*(P;z)}{(1-z)^{d+1}}, \qquad h^*(P;z)=\sum_{i=0}^{s}h_i^*z^i, \qquad s=\deg h^*(P;z). \] The coefficient sequence is unimodal if there is an index $k\in\{0,\ldots,s\}$ such that \[ h_0^*\leq\cdots\leq h_k^*\geq\cdots\geq h_s^*. \] It is log-concave if \[ (h_i^*)^2\geq h_{i-1}^*h_{i+1}^* \qquad(1\leq i\leq s-1). \] These definitions include the cases of degree zero or one, with empty families of inequalities interpreted vacuously.

Write $L^*=\operatorname{Hom}_{{\mathbb{Z}}}(L,{\mathbb{Z}})$. A full-dimensional lattice polytope $Q\subseteq L_{{\mathbb{R}}}$ containing the origin in its interior is reflexive if its polar \[ Q^{\vee}=\{u\in L^*\otimes_{{\mathbb{Z}}}{\mathbb{R}}:\langle u,x\rangle\leq 1\text{ for every }x\in Q\} \] is a lattice polytope in the dual lattice $L^*$. The polytope $P$ is Gorenstein if there exist an integer $r\geq 1$ and a lattice point $a\in\operatorname{int}(rP)\cap L$ such that $a$ is the unique interior lattice point of $rP$ and $rP-a$ is reflexive.

The source asks whether each of the following universally quantified assertions is true, over all such lattices and lattice polytopes, without a dimension bound:

1. If $P$ has IDP, then $h^*(P;z)$ is unimodal.
2. If $P$ has IDP, then $h^*(P;z)$ is log-concave.
3. If $P$ is Gorenstein and has IDP, then $h^*(P;z)$ is unimodal.
4. If $P$ is Gorenstein and has IDP, then $h^*(P;z)$ is log-concave.

These are four separate questions. The supplied dated curation reports that assertion (3) is proved, assertion (1) remains open, and assertion (2) has counterexamples. It does not explicitly determine assertion (4). These are supplied literature assessments, not independently verified conclusions of this draft.

Short English statement

The problem asks four questions about the coefficients of the Ehrhart $h^*$-polynomial: whether integer decomposition forces unimodality or log-concavity, and whether adding the Gorenstein condition forces either property. The supplied record labels the compound problem partially solved; its dated assessment reports a proof for Gorenstein IDP unimodality, counterexamples to general IDP log-concavity, and an open general IDP unimodality question. It gives no explicit separate status for Gorenstein IDP log-concavity.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003607$, title Unimodality and Log-Concavity under Integer Decomposition, in the Oberwolfach Reports Open Problems collection; exact source problem code catalogue entry. No individual proposer is identified. The supplied proposed year is 2017.
• Supplied original-report citation: Mini-Workshop: Lattice Polytopes: Methods, Advances, Applications (2018), Oberwolfach Reports, pp. 2659--2701, DOI 10.4171/owr/2017/44. Supplied source URL: https://doi.org/10.4171/owr/2017/44. The citation year is retained as supplied despite the different year in the DOI and proposed-year field.
• Supplied literature source: Adiprasito et al., beyond positivity in Ehrhart theory, https://arxiv.org/abs/2210.10734. The supplied assessment attributes the proof of Gorenstein IDP $h^*$-unimodality to Adiprasito--Papadakis--Petrotou--Steinmeyer.
• Supplied literature source: Hofscheier--Kurylenko--Nill, non-log-concave IDP examples, https://arxiv.org/abs/2505.18896.
• Supplied literature source labeled original OWR report: https://doi.org/10.4171/OWR/2017/44.
• Literature checked at: 2026-08-21. The partially solved label and the accompanying assessment are supplied dated curation, not independently verified here. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The record supplies the statement-status label corrected_verified and reports manual semantic, typographic, and LaTeX review. That source-side review is distinct from mathematical review of this statement-only draft.