Spanning Rank of Lattice Polytopes

Definitions and mathematical statement

The source uses $\operatorname{aff}(S)={\mathbb{Z}}^d$ without specifying the meaning of $\operatorname{aff}$. It cannot mean the usual real affine hull: for $d\geq 1$, a real affine subspace cannot equal ${\mathbb{Z}}^d$. The following formulation is therefore conditional on interpreting it as the integer affine span.

For a nonempty finite set $S\subseteq{\mathbb{Z}}^d$, define its integer affine span by \[ \operatorname{aff}_{{\mathbb{Z}}}(S)=\left\{\sum_{s\in S} n_s s : n_s\in{\mathbb{Z}},\ \sum_{s\in S}n_s=1\right\}. \] For each positive integer $d$, let $\mathcal P_d$ be the collection of full-dimensional lattice polytopes $P\subseteq{\mathbb{R}}^d$, meaning convex hulls of finite subsets of ${\mathbb{Z}}^d$ with real affine dimension $d$, such that \[ \operatorname{aff}_{{\mathbb{Z}}}(P\cap{\mathbb{Z}}^d)={\mathbb{Z}}^d. \] For $P\in\mathcal P_d$, set \[ \operatorname{SR}(P)=\min\left\{|B|:B\subseteq P\cap{\mathbb{Z}}^d,\ \operatorname{aff}_{{\mathbb{Z}}}(B)={\mathbb{Z}}^d\right\}, \qquad \operatorname{SR}(d)=\max_{P\in\mathcal P_d}\operatorname{SR}(P). \] The supplied record reports a finite bound on $\operatorname{SR}(P)$ depending only on $d$; this reported bound would ensure that the displayed maximum exists. The original statement supplies the values \[ \operatorname{SR}(1)=2,\qquad \operatorname{SR}(2)=3,\qquad \operatorname{SR}(3)=5. \] Subject to confirmation of the integer affine span interpretation, determine $\operatorname{SR}(4)$ and the asymptotic order of $\operatorname{SR}(d)$ as $d\to\infty$. The bound, the three stated values, and the open status are supplied source claims, not independently verified assertions in this draft.

Short English statement

Under the explicitly conditional interpretation of affine span as integer affine span, find the worst-case minimum number of lattice points needed to generate the entire ambient affine lattice from a spanning lattice polytope. Determine this number in dimension four and its asymptotic order as the dimension grows. The source's span notation requires confirmation.

Sources

• Local UnsolvedMath registry: permanent ID $30004236$, title Spanning Rank of Lattice Polytopes, collection Oberwolfach Reports Open Problems, exact source code catalogue entry. The record supplies proposed year 2019 and names no individual proposer.
• Supplied report citation: Geometric, Algebraic, and Topological Combinatorics (2020), Oberwolfach Reports, pp. 2395--2472, DOI 10.4171/owr/2019/39. https://doi.org/10.4171/owr/2019/39. The background describes the report as documenting workshop open-problem discussions.
• Supplied literature reference attributed to Averkov, Hofscheier, and Nill: https://arxiv.org/abs/1911.03511. The record provides no paper title or pinpoint reference.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports dimension-only boundedness and known values in dimensions one through three, and describes the dimension-four value and asymptotic order as unresolved. This is supplied dated curation, not an independently verified literature assessment. The record itself calls for independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and reports that the official Oberwolfach report was checked directly. These are supplied provenance claims; no source was independently visited or checked for this draft.