Bondal-Thomsen Polytopes within Lattice-Polytope Theory

Definitions and mathematical statement

Let $M$ be a lattice, meaning a free abelian group of finite rank, and let $M_{{\mathbb{R}}}=M\otimes_{{\mathbb{Z}}}{\mathbb{R}}$. A lattice polytope in $M_{{\mathbb{R}}}$ is the convex hull of a finite nonempty subset of $M$. A rational polytope is the convex hull of a finite nonempty subset of $M\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$. For a rational polytope $P$, its denominator relative to $M$ is \[ \operatorname{den}_{M}(P)=\min\{d\in{\mathbb{Z}}_{>0}:dP\text{ is a lattice polytope relative to }M\}. \] A zonotope is a Minkowski sum of finitely many line segments: \[ Z=[a_1,b_1]+\cdots+[a_r,b_r],\qquad [a,b]=\{(1-t)a+tb:0\leq t\leq1\}. \] It is a lattice zonotope when such a representation exists with all endpoints in $M$.

Source problem, with unresolved specifications. The supplied record refers to a finite collection of Bondal--Thomsen polytopes. Denote that referenced collection provisionally by $\mathcal B$. The requested research task is to characterize its intrinsic combinatorial properties, determine denominator bounds relative to the intended ambient lattices, and describe its relationships with zonotopes and toric arrangements, thereby placing it within the general theory of lattice polytopes.

The record does not construct $\mathcal B$, specify its ambient lattices or underlying geometric data, or establish that its members are lattice or rational polytopes in those lattices. The definitions above therefore provide terminology only; they do not assert these properties for $\mathcal B$. The scope of the requested denominator bounds and the intended meaning of the relationships with toric arrangements are also unspecified. Consequently, the supplied material determines a research question but does not determine a fully quantified conjecture or a precise classification target.

Short English statement

The question asks how the referenced finite collection of Bondal--Thomsen polytopes can be characterized using lattice-polytope theory, including denominators and connections with zonotopes and toric arrangements. The collection and its geometric hypotheses are missing, so a fully precise formulation requires further source context. The supplied status is partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30006304$, title Bondal--Thomsen Polytopes within Lattice-Polytope Theory, source problem code catalogue entry. The record gives proposed year 2025 and names no proposer.
• Supplied report citation: Toric Geometry (2025), Oberwolfach Reports, DOI 10.4171/owr/2025/19, pp. 883--940. https://doi.org/10.4171/owr/2025/19. No problem-specific page or complete contextual passage is supplied.
• Supplied literature reference: Bauermeister et al., Strata of toric hyperplane arrangements. https://arxiv.org/abs/2507.14356.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved, states that the broad lattice-polytope question remains open, attributes denominator bounds to the report, and attributes further zonotope and lattice-point structure related to the Bondal--Thomsen collection to the 2025 toric-arrangement paper. This is supplied dated curation, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified, explaining that only formula delimiters, notation, grammar, and typography were repaired and that mathematical content and scope were preserved. This supplied editorial label does not establish mathematical correctness or resolve the missing context.