Semicircle Packings and Colorings of Simplicial 4-Polytopes

Definitions and mathematical statement

The record contains two separate questions. Several conventions needed for a fully precise statement are not specified; the formulations below make those conventions explicit and remain conditional on their confirmation.

For a plane curve $\gamma\subset{\mathbb{R}}^2$ and $r>0$, put \[ B_r=\{x\in{\mathbb{R}}^2:\|x\|\le r\},\qquad K_{r,\gamma}=\gamma+B_r=\{x+y:x\in\gamma,\ y\in B_r\}. \] The source denotes by $f_c(r,\gamma)$ the maximum density of a packing by congruent copies of $K_{r,\gamma}$. It does not specify the density convention or establish that a maximum is attained. One conventional precise interpretation is \[ f_c(r,\gamma)=\sup_{\mathcal P}\limsup_{T\to\infty}\frac{\operatorname{area}\!\left(B_T\cap\bigcup_{K\in\mathcal P}K\right)}{\operatorname{area}(B_T)}, \] where $\mathcal P$ ranges over locally finite families of congruent copies of $K_{r,\gamma}$ with pairwise disjoint interiors. Here congruence is provisionally taken to allow all Euclidean isometries; these packing and density conventions require confirmation against the source.

Semicircle packing question. Interpreting the semicircle as a fixed closed circular arc, write \[ \gamma_1=\{R(\cos t,\sin t):0\le t\le\pi\},\qquad R>0\text{ fixed}. \] The intended question appears to be whether \[ f_c(r,\gamma_1)=\Theta(\sqrt r)\qquad(r\downarrow0). \] Under this interpretation, the assertion asks whether there exist constants $c,C,r_0>0$, independent of $r$ but allowed to depend on the fixed semicircle, such that \[ \forall r\in(0,r_0),\qquad c\sqrt r\le f_c(r,\gamma_1)\le C\sqrt r. \] The supplied statement does not explicitly give the limiting regime, the radius normalization, or the endpoint convention. Thus $r\downarrow0$ and the displayed arc are explicit provisional interpretations, not additional verified source facts.

Polytope coloring question. A simplicial convex $4$-polytope is a convex polytope of affine dimension $4$ whose facets are tetrahedra; its $2$-faces are triangles. Under the intended but unstated interpretation that vertices are colored, the question is whether \[ \exists k\in{\mathbb{N}},\ k\ge1,\quad\forall P\text{ a simplicial convex }4\text{-polytope},\quad\exists\chi:V(P)\to\{1,\ldots,k\} \] such that, for every $2$-face $F$ of $P$, \[ \left|\{\chi(v):v\in V(F)\}\right|\ge2. \] Thus one constant $k$ must work for every such polytope, regardless of its number of vertices. No proper coloring of the edges is required by this condition. The vertex-coloring interpretation and the ordinary convex meaning of polytope require confirmation because the supplied wording does not expressly state them.

Short English statement

The first question asks whether the best packing density of a thin circular semicircle grows on the order of $\sqrt r$, provisionally as $r\downarrow0$. The second asks whether a universal finite number of vertex colors suffices for every simplicial convex $4$-polytope so that no triangular face has all vertices the same color. The limiting regime, packing conventions, and coloring domain need source confirmation.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30001898$, source problem code catalogue entry, title Semicircle Packings and Colorings of Simplicial 4-Polytopes; collection: Oberwolfach Reports Open Problems.
• Supplied source citation: Discrete Geometry (2012), Oberwolfach Reports, pp. 2459--2548, DOI 10.4171/owr/2011/44. Supplied source URL: https://doi.org/10.4171/owr/2011/44. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference, labeled OWR 44 (2012): https://doi.org/10.4171/owr/2011/44. Literature assessment dated exactly 2026-08-21: the record labels the problems open and states that the semicircle packing asymptotic and bounded-coloring question remain posed without a complete general theorem. This is supplied dated curation derived from the OWR record, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• Attribution: no individual proposer is supplied. The registry gives proposed year $2011$ and the tag owr-year:2011, while its report title and literature label use $2012$; these supplied date labels are retained without reconciliation.
• The registry labels the statement corrected_verified and reports that the official report was checked and the packing-density function, semicircle case, and bounded-color polytope question were restored. This is a supplied verification claim, not verification performed for this draft.