Maximum Volume of Canonical Fano Polytopes

Definitions and mathematical statement

Let $d\in{\mathbb{Z}}$ with $d\geq4$, let $N$ be a lattice of rank $d$, and put $N_{{\mathbb{R}}}=N\otimes_{{\mathbb{Z}}}{\mathbb{R}}$. A $d$-dimensional lattice polytope is the convex hull of finitely many points of $N$, with affine span $N_{{\mathbb{R}}}$. Such a polytope $P$ is called canonical Fano if \[ \operatorname{int}(P)\cap N=\{0\}. \] Use lattice-relative ordinary volume $\operatorname{vol}_N$, for which a fundamental parallelepiped of $N$ has volume $1$. The supplied record does not explicitly specify its volume normalization; this convention, consistent with the displayed denominator $d!$, requires confirmation against the source.

Define the Sylvester sequence by \[ s_1=2,\qquad s_{i+1}=\prod_{j=1}^{i}s_j+1\quad(i\geq1). \] The proposed assertion is that every such $P$ satisfies \[ \operatorname{vol}_N(P)\leq\frac{2(s_d-1)^2}{d!}, \] with equality only for the canonical Fano simplex denoted $R_{(d)}$ in the supplied statement (written $R(d)$ in its original-statement field).

The equality clause remains incompletely specified: the supplied record gives no definition or vertices of $R_{(d)}$ and does not specify the equivalence relation intended by "only for." Thus the inequality is formulated under the stated volume convention, while a precise identification of its equality cases requires additional source information. The source asks for a sharp bound and equality characterization; these are proposed assertions, not conclusions established here.

Short English statement

For every integer $d\geq4$, the problem asks whether a $d$-dimensional lattice polytope whose unique interior lattice point is the origin has volume at most $2(s_d-1)^2/d!$, with equality only for a designated simplex $R_{(d)}$. The record does not define that simplex or the intended equivalence of equality cases, and its volume convention needs confirmation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003597$, title Maximum Volume of Canonical Fano Polytopes, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2017.
• 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.
• Supplied literature reference: Balletti--Kasprzyk--Nill, "maximum dual volume," https://doi.org/10.1017/fms.2022.93.
• Supplied literature reference: Averkov--Krumpelmann--Nill, "largest integral simplices," https://arxiv.org/abs/1309.7967.
• Supplied literature reference: "original OWR report," https://doi.org/10.4171/OWR/2017/44.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the canonical Fano volume bound is proved for canonical Fano simplices, that the dual-volume analogue is proved in general, and that the sharp primal-volume bound and equality characterization remain open for general canonical Fano polytopes. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment; its status and cited claims require independent verification.