Grassmann Polytopes as Positive Geometries

Definitions and mathematical statement

For integers $1\leq k\leq n$, let $\operatorname{Gr}(k,n)({\mathbb{R}})$ denote the Grassmannian of $k$-dimensional linear subspaces of ${\mathbb{R}}^n$. Its totally nonnegative part is \[ \operatorname{Gr}_{\geq 0}(k,n)=\{V\in\operatorname{Gr}(k,n)({\mathbb{R}}):\text{the Pl\"ucker coordinates of }V\text{ admit a common representative with all coordinates nonnegative}\}. \] Strata of this set can be specified by requiring designated Plücker coordinates to vanish and the remaining coordinates to be positive. For a linear map $Z:{\mathbb{R}}^n\to{\mathbb{R}}^r$, the induced map \[ V\longmapsto Z(V)\in\operatorname{Gr}(k,r)({\mathbb{R}}) \] is defined only where $\dim Z(V)=k$. Images of totally nonnegative Grassmannians or their strata under such maps provide the relevant Grassmann-polytope constructions. The supplied record does not specify the permitted parameters, strata, hypotheses on $Z$, or image and closure conventions; these must be recovered from the cited sources before fixing the intended class.

A positive geometry consists of a complex projective algebraic variety $X$, defined over ${\mathbb{R}}$, together with an appropriate oriented real semialgebraic region $P\subseteq X({\mathbb{R}})$. Its defining canonical-form requirement is the existence of a unique nonzero rational differential form $\Omega(X,P)$ of degree $\dim X$, with only logarithmic poles along the algebraic boundary and no poles elsewhere, whose residues on boundary components are the canonical forms of the induced lower-dimensional positive geometries. The recursion ends at oriented points, with canonical form $+1$ or $-1$. Applying this definition requires specification of the ambient variety, orientation, and boundary geometries, not merely the underlying point set.

Question as supplied. Is every Grassmann polytope a positive geometry, including the amplituhedra singled out in the supplied background? Determine the topology and combinatorics of Grassmann polytopes in a form analogous to the theory of ordinary convex polytopes.

More explicitly, if $\mathcal G$ denotes the intended class of Grassmann polytopes with their intended ambient varieties $X_P$, the first question asks whether \[ \forall P\in\mathcal G,\qquad (X_P,P)\text{ is a positive geometry}. \] This is a formulation template: the source record does not determine $\mathcal G$, $X_P$, or all required boundary conventions precisely enough to turn it into a fully specified universal assertion. The topology and combinatorics request is likewise a research objective; the record does not specify particular invariants, a face relation, or a proposed classification.

Short English statement

The supplied problem asks whether every Grassmann polytope has the recursive canonical-form structure of a positive geometry, and seeks a theory of its topology and combinatorics analogous to convex-polytope theory. The intended construction and boundary conventions need clarification. The supplied curation labels the general assertion open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004215$, title Grassmann Polytopes as Positive Geometries, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. The record gives proposed year 2019 and does not name a 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 literature-source list labels this reference "Oberwolfach Report 39, Grassmann polytopes."
• Supplied literature reference: Arkani-Hamed, Bai, and Lam, positive geometries. https://doi.org/10.1007/JHEP11%282017%29039.
• Literature assessment checked at 2026-08-22: the supplied dated curation labels the general positive-geometry assertion open, particularly for amplituhedra, and states that special cases and canonical-form results do not establish the general claim. This is supplied dated curation, not an independently verified literature assessment; the record explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, and sentence structure were repaired. That supplied editorial label does not constitute mathematical verification of this draft.