Convex Geometry of General Polytope Face Vectors

Definitions and mathematical statement

For each integer $d\geq4$, interpret $\mathcal P^d$ conventionally as the class of convex polytopes of dimension $d$. For $P\in\mathcal P^d$, let $f_i(P)$ denote the number of its $i$-dimensional faces, and set \[ f(P)=(f_0(P),\ldots,f_{d-1}(P))\in{\mathbb{R}}^d, \qquad S_d=\{f(P):P\in\mathcal P^d\}. \] Let $\sigma^d$ be a $d$-dimensional simplex and write \[ a_d=f(\sigma^d),\qquad (a_d)_i=\binom{d+1}{i+1}\quad(0\leq i\leq d-1). \] These are conventional interpretations of notation that the supplied record does not explicitly define.

For $S\subseteq{\mathbb{R}}^d$, define its convex hull by finite convex combinations: \[ \operatorname{Conv}(S)=\left\{\sum_{j=1}^m\lambda_js_j:m\geq1,\ s_j\in S,\ \lambda_j\geq0,\ \sum_{j=1}^m\lambda_j=1\right\}. \] One conventional interpretation of the cone with apex $a_d$ is \[ C_d=\operatorname{Cone}_{a_d}(S_d) =a_d+\left\{\sum_{j=1}^m t_j(s_j-a_d):m\geq1,\ s_j\in S_d,\ t_j\geq0\right\}. \] No topological closure is included in this displayed definition. The source does not specify whether its cone notation includes closure, so the following formulation is conditional on this convention.

Questions. For every integer $d\geq4$, does \[ \operatorname{Conv}(S_d)=C_d \] hold? For each such $d$, is $C_d$ finitely generated, meaning that there exist an integer $r\geq1$ and vectors $v_1,\ldots,v_r\in{\mathbb{R}}^d$ such that \[ C_d=a_d+\left\{\sum_{j=1}^r t_jv_j:t_j\geq0\right\}? \] The source additionally asks whether the corresponding rays are dense and which polytopes determine, or lie on, the boundary of this cone. These two requests do not yet specify unique mathematical questions. In particular, the source does not identify the family of rays, the space in which density is intended, or its topology. Possible face-vector rays would have the form \[ a_d+{\mathbb{R}}_{\geq0}\left(f(P)-a_d\right),\qquad f(P)\neq a_d, \] but this family and any density target require confirmation. Likewise, a boundary question must specify whether boundary is taken in ${\mathbb{R}}^d$, relative to $\operatorname{aff}(C_d)$, or in a space of directions. It must also distinguish characterizing polytopes with boundary face vectors from characterizing generators or supporting faces of the cone. No choice among these meanings is asserted here.

Short English statement

For each dimension $d\geq4$, ask whether the convex hull of polytope face vectors equals the cone based at the simplex face vector, and whether that cone has finitely many generators. The source also asks about density of rays and boundary polytopes, but leaves the relevant rays, topology, and boundary convention unspecified. The definitions above therefore give a conditional statement requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004224$, title Convex Geometry of General Polytope Face Vectors, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2019.
• 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 record describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labeled "Ziegler on f-vectors": https://doi.org/10.1515/advgeom-2018-0025. No fuller bibliographic citation or precise supporting result is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the report leaves the convex-hull, finite-generation, density, and boundary questions open for general $d\geq4$, while later work gives additive structure without resolving the stated equalities. 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.
• The registry labels the statement corrected_verified and says that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This supplied editorial label does not establish mathematical correctness or resolve the formulation issues recorded here.