Centrally Symmetric Two-Level Reflexive Polytopes

Definitions and mathematical statement

Use the standard lattice-polytope conventions as follows. Let $d\geq 1$, let $M$ be a lattice of rank $d$, let $M_{{\mathbb{R}}}=M\otimes_{{\mathbb{Z}}}{\mathbb{R}}$, and let $N=\operatorname{Hom}_{{\mathbb{Z}}}(M,{\mathbb{Z}})$ be the dual lattice. A full-dimensional lattice polytope $P\subset M_{{\mathbb{R}}}$ is the convex hull of finitely many points of $M$ and satisfies $\dim P=d$. Write $V(P)$ for its vertex set. The polytope is centrally symmetric if there is $c\in M_{{\mathbb{R}}}$ such that $P=2c-P$.

The polytope $P$ is reflexive if $0\in\operatorname{int}(P)$ and its polar \[ P^{\vee}=\{y\in N\otimes_{{\mathbb{Z}}}{\mathbb{R}}:\langle y,x\rangle\leq 1\text{ for every }x\in P\} \] is a lattice polytope with respect to $N$. It is $2$-level if, for every facet $F$ and a supporting equation $\ell_F(x)=b_F$, chosen with $P\subseteq\{x:\ell_F(x)\leq b_F\}$, there is $a_F<b_F$ such that \[ \{\ell_F(v):v\in V(P)\}=\{a_F,b_F\}. \] Thus every vertex lies on the facet hyperplane or on one other parallel hyperplane.

Classification problem. For arbitrary positive dimension $d$, characterize all full-dimensional lattice polytopes $P\subset M_{{\mathbb{R}}}$ that are simultaneously centrally symmetric, $2$-level, and reflexive. A characterization should give necessary and sufficient conditions for membership in this class, or an exhaustive description with an explicitly stated equivalence relation.

Extension question and unresolved qualifications. The supplied source additionally asks whether the characterization of its "exhibited examples" extends to larger classes of $2$-level polytopes, or to all $2$-level polytopes. Neither those examples nor their characterization is included in the record, so this extension cannot be formulated as a specific mathematical assertion from the supplied evidence. The record also does not specify the equivalence relation for classification, whether the larger classes must remain lattice polytopes, or which hypotheses are to be retained in those classes. The definitions above use standard full-dimensional, positive-dimensional lattice conventions; the source does not explicitly address dimension-zero cases or relative-dimensional formulations.

Short English statement

Classify centrally symmetric reflexive lattice polytopes whose vertices occupy exactly two parallel levels relative to each facet. The proposed extension to broader classes of $2$-level polytopes requires the omitted examples and their characterization. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003605$, source problem code catalogue entry, title Centrally Symmetric Two-Level Reflexive Polytopes, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied. The record gives proposed year 2017 and a supplied statement-review label corrected_verified; these are registry assertions, not independent verification here.
• Supplied 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 background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied 2018 report designation and 2017 proposed-year metadata have not been reconciled.
• Supplied literature source, "original OWR report": https://doi.org/10.4171/OWR/2017/44.
• Supplied literature source, "Nill, pseudo-symmetric simplicial reflexive polytopes": https://arxiv.org/abs/math/0511294.
• Supplied literature source, "Castillo--Liu, generalized permutohedra": https://arxiv.org/abs/1509.07884.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the classification and its proposed broader extensions open, and states that known examples and narrower simplicial classifications do not provide a general classification. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.