Volume Bounds for Clean Lattice Tetrahedra

Definitions and mathematical statement

A lattice tetrahedron is a full-dimensional polytope \[ T=\operatorname{conv}\{v_0,v_1,v_2,v_3\}\subset{\mathbb{R}}^3, \qquad v_0,v_1,v_2,v_3\in{\mathbb{Z}}^3. \] Using the standard interpretation of clean, require \[ \partial T\cap{\mathbb{Z}}^3=\{v_0,v_1,v_2,v_3\}. \] Write $I(T)=\operatorname{int}(T)\cap{\mathbb{Z}}^3$, and let $\mathcal C_k$ be the class of clean lattice tetrahedra with $\#I(T)=k$, for each integer $k\geq1$. The interior lattice points are collinear if there is an affine line $L\subset{\mathbb{R}}^3$ with $I(T)\subset L$. In particular, sets of one or two interior points are collinear. Let $V(T)$ denote ordinary Euclidean volume; normalized lattice volume is $6V(T)$.

The supplied question refers to Duong's volume conjecture but does not state its numerical bound or its equality conditions. If $B(k)$ denotes its intended conjectured maximum, expressed in Euclidean-volume units, the volume-bound questions have the following schematic form: \[ \begin{aligned} &\forall k\in{\mathbb{Z}}_{\geq1}\ \forall T\in\mathcal C_k, &&I(T)\text{ collinear}\ \Longrightarrow\ V(T)\leq B(k),\\ &\forall k\in{\mathbb{Z}}_{\geq1}\ \forall T\in\mathcal C_k, &&I(T)\text{ not collinear}\ \Longrightarrow\ V(T)<B(k). \end{aligned} \] Here $B(k)$ is an unspecified target from the source, not a newly defined or established bound. Consequently, these formulas do not constitute a complete numerical formulation of Duong's conjecture. The record also leaves unclear whether "the maximum" means the conjectured bound or an independently defined attained extremal volume. Its background mentions a full uniqueness question, but supplies neither an extremal tetrahedron nor the equivalence relation under which uniqueness is intended.

The supplied literature assessment labels the problem partially solved and reports a proof for the collinear case, with the non-collinear case and full uniqueness question unresolved. These are supplied dated curation claims, not independently verified mathematical conclusions.

Short English statement

For each positive integer $k$, consider clean lattice tetrahedra with exactly $k$ interior lattice points. The source asks whether Duong's conjectured volume bound holds when those points are collinear, and whether non-collinearity forces strict inequality. The conjectured bound and uniqueness formulation are missing. Supplied dated curation reports the collinear case as proved.

Sources

• Local UnsolvedMath registry, permanent ID $30003604$, title: Volume Bounds for Clean Lattice Tetrahedra; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. This draft uses the supplied complete registry record.
• Supplied source 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 record gives proposed year 2017 and names Duong's conjecture, but identifies no proposer of this workshop question.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/OWR/2017/44.
• Supplied literature source, labeled "Balletti--Kasprzyk, three-dimensional polytopes with interior points": https://arxiv.org/1612.08918.
• Supplied literature source, labeled "Averkov--Krumpelmann--Nill, volume bounds": https://arxiv.org/1309.7967.
• Literature checked at: 2026-08-21. This is supplied dated curation derived from the OWR record, not an independently verified assessment. It reports the collinear case as proved and the non-collinear case and full uniqueness question as unresolved; the record explicitly requests independent verification of time-sensitive status and cited claims.