Polyhedrality of Totally Nonnegative Critical Varieties

Definitions and mathematical statement

The supplied source denotes a family of totally nonnegative critical varieties by $\operatorname{Crit}^{\geq 0}_f$. It does not specify the admissible indices $f$, the construction of these varieties, their ambient spaces, or the meaning of total nonnegativity in this setting. These are source-dependent definitions that must be recovered before a fully specified mathematical statement can be given.

Question, retaining the source's unresolved terminology. For every admissible index $f$ in the source's family, is $\operatorname{Crit}^{\geq 0}_f$ itself a polytope?

In standard real convex geometry, a nonempty polytope is a set of the form \[ P=\operatorname{conv}\{v_1,\ldots,v_m\},\qquad m\geq 1, \] where the points lie in a finite-dimensional real affine space. The supplied record does not specify an affine realization of $\operatorname{Crit}^{\geq 0}_f$, or whether the intended assertion concerns a geometric realization, a homeomorphism, or another specified equivalence with a polytope. These interpretations are not interchangeable, so this draft does not select one.

The supplied dated assessment describes each such variety as an image of an affine poset cyclohedron and identifies a top-dimensional case with a polytope. It also refers to every critical cell, without specifying its relation to the varieties in the question. These are supplied background qualifications, not additional hypotheses, established results of this draft, or a restriction of the universal question.

Short English statement

The question asks whether every totally nonnegative critical variety $\operatorname{Crit}^{\geq 0}_f$ is itself a polytope. The supplied record labels the problem open, but omits the defining construction, the range of $f$, and the precise meaning of being a polytope in this setting. A fully specified formulation requires human review of those source-dependent details.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004938$, title Polyhedrality of Totally Nonnegative Critical Varieties, exact source problem code catalogue entry. No individual proposer is identified.
• Mini-Workshop: Scattering Amplitudes, Cluster Algebras, and Positive Geometries (2022), Oberwolfach Reports, pp. 3101--3133, DOI 10.4171/owr/2021/57: https://doi.org/10.4171/owr/2021/57. The extracted problem passage identifies p. 3107. The registry separately records proposed year 2021.
• Supplied literature reference: Galashin, totally nonnegative critical varieties, https://arxiv.org/abs/2110.08548.
• Supplied literature reference labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2021/57.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, describes an affine poset cyclohedron image representation for each variety and a polytope identification in the top-dimensional case, and says the assertion for every critical cell remains unproved. This is supplied dated curation, not independently verified here; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports prior direct checking of the official report. That verification description is supplied provenance, not verification performed for this draft.