Contractibility of Convex-Position Polygon Spaces

Definitions and mathematical statement

The supplied record defines a $k$-gon to be a convex polygon with at most $k$ vertices. An ordered arrangement is a tuple $(P_1,\ldots,P_n)$, with the labels retained when taking equivalence classes. The source asks whether the space of these arrangements in convex position, modulo affine transformations, is contractible.

A conditional formalization is as follows. If the intended ambient space is ${\mathbb{R}}^2$ and polygons are compact convex sets with nonempty interior, put \[ \mathcal P_{\leq k}=\{P\subseteq{\mathbb{R}}^2:P\text{ is a compact convex polygon with nonempty interior and at most }k\text{ vertices}\}, \] and let \[ X_{n,k}=\{(P_1,\ldots,P_n)\in\mathcal P_{\leq k}^{\,n}:(P_1,\ldots,P_n)\text{ is in convex position}\}. \] Here "in convex position" remains an unresolved source condition: the supplied record does not define it. If all invertible affine transformations of the plane are intended, let \[ G=\operatorname{Aff}({\mathbb{R}}^2)=\{x\mapsto Ax+b:A\in\operatorname{GL}_2({\mathbb{R}}),\ b\in{\mathbb{R}}^2\} \] act simultaneously on the polygons, without permuting their labels. One possible topology on $X_{n,k}$ is the subspace topology induced by the product Hausdorff metric on compact sets, where \[ d_H(P,Q)=\max\left\{\sup_{p\in P}\inf_{q\in Q}\|p-q\|,\ \sup_{q\in Q}\inf_{p\in P}\|q-p\|\right\}. \] Under this interpretation, give $Y_{n,k}=X_{n,k}/G$ the quotient topology. The intended question would then be whether, for every admissible pair $(n,k)$, there exist $y_0\in Y_{n,k}$ and a continuous map \[ H:Y_{n,k}\times[0,1]\longrightarrow Y_{n,k},\qquad H(y,0)=y,\quad H(y,1)=y_0\quad(y\in Y_{n,k}). \] This is a conditional formulation template. The supplied record does not specify the definition of convex position, the topology, admissible parameter ranges, or degeneracy conventions. In particular, the planar, nonempty-interior and Hausdorff-topology choices above are explicitly provisional. The usual nondegenerate planar range would be $n\geq1$ and $k\geq3$, but this range is not stated in the record. These qualifications must be resolved before the template can be treated as a precise source-faithful statement.

Short English statement

The question asks whether the space of labelled arrangements of $n$ convex polygons, each with at most $k$ vertices and satisfying the intended convex-position condition, can be continuously contracted to a point after identifying arrangements related by a common affine transformation. The supplied record leaves essential conventions undefined, so the displayed formulation is conditional and requires mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30002670$, title "Contractibility of Convex-Position Polygon Spaces," source problem code catalogue entry, collection "Oberwolfach Reports Open Problems." No individual proposer is supplied; the proposed year is recorded as $2014$.
• Supplied source citation: Discrete Geometry (2015), Oberwolfach Reports, DOI 10.4171/owr/2014/40, pp. 2235--2296. https://doi.org/10.4171/owr/2014/40. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The citation's year $2015$ differs from the recorded proposed year $2014$ and the year appearing in the DOI; this discrepancy is retained without resolution.
• Supplied literature reference: Dobbins--Holmsen--Hubard, "convex-body realization spaces." https://doi.org/10.1007/s00454-017-9869-8.
• Supplied literature reference: "Oberwolfach report." https://ems.press/content/serial-article-files/46530.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open and states that realization spaces of unrestricted convex-body arrangements are understood, that bounded-vertex polygon realization spaces can have complicated topology, and that the convex-position $k$-gon contractibility question remains open. These are supplied dated curation claims, not independently verified conclusions. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report passage was checked directly and its question, definitions and notation restored without strengthening the claim. This is supplied verification metadata; no independent source inspection or mathematical verification is asserted here. The original wording omits "ordered," whereas the cleaned statement and statement field explicitly include it.