Polyhedrality of Cones of Nonnegative Valuations

Definitions and mathematical statement

Write $C=\overline{V}_+$ for the cone called the cone of lattice-invariant $h^*$-nonnegative valuations in the supplied record. The record does not define the domain or codomain of these valuations, the invariance group, the relevant $h^*$-coefficients, or the meaning of the overline. These definitions must be recovered before a self-contained conjecture can be stated.

For a finite-dimensional real vector space $W$, a cone $C\subseteq W$ is polyhedral if there are finitely many linear functionals $\ell_1,\ldots,\ell_m\in W^*$ such that \[ C=\{w\in W:\ell_i(w)\geq 0\text{ for }1\leq i\leq m\}. \] For a nonnegative integer $d$, define \[ {\mathbb{R}}_{\geq 0}^{d+1}=\{(x_0,\ldots,x_d)\in{\mathbb{R}}^{d+1}:x_i\geq 0\text{ for every }i\}. \] A cone $C$ in a real vector space is linearly isomorphic to this orthant if there is a linear bijection \[ T:\operatorname{span}_{{\mathbb{R}}}(C)\longrightarrow{\mathbb{R}}^{d+1} \quad\text{with}\quad T(C)={\mathbb{R}}_{\geq 0}^{d+1}. \] Subject to identifying the intended cone, its ambient space, and the dimension parameter $d$, the supplied question is: is $\overline{V}_+$ polyhedral? In the stated stronger form, is $\overline{V}_+$ linearly isomorphic to ${\mathbb{R}}_{\geq 0}^{d+1}$? The source does not specify the range of $d$, the dimension quantifier, or whether the intended ambient space is finite-dimensional.

The supplied background reports verification in dimensions at most three, but its dated assessment says that the underlying conjecture and its strong version were not identified. Consequently, this reported verification cannot be assigned confidently to either formulation above.

Short English statement

The question asks whether the specified cone of valuations is polyhedral and, more strongly, whether it is a linear copy of a nonnegative orthant. Essential definitions and dimension quantifiers are missing. The supplied open label and low-dimensional verification report require review.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30002779$, title "Polyhedrality of Cones of Nonnegative Valuations," exact source problem code catalogue entry. The record assigns the proposed year 2015 and supplies no individual proposer.
• Supplied report citation: Geometric and Algebraic Combinatorics (2015), Oberwolfach Reports, pp. 285--368, DOI 10.4171/owr/2015/5. Supplied source URL and literature source labelled "original OWR report": https://doi.org/10.4171/owr/2015/5. The record attributes its collection to workshop open-problem discussions documented in Oberwolfach Reports; no problem-specific page is supplied.
• Literature assessment checked at 2026-08-21: the supplied status is "open." The supplied assessment says that verification in dimensions at most three is reported, but that the underlying conjecture and its strong version are unidentified, leaving the extracted claim unresolved as stated. This is supplied dated curation, not an independently verified literature assessment; the background requests independent verification of time-sensitive status and cited claims.
• The registry separately labels the statement corrected_verified and says that notation and publication prose were repaired without changing its substance. This is supplied editorial metadata, not independent mathematical verification, and it does not resolve the missing definitions identified above.