Regular Octahedra Inscribed in Smooth Two-Spheres

Definitions and mathematical statement

Let $S^2=\{x\in{\mathbb{R}}^3:\|x\|=1\}$, with its usual smooth structure. A $C^\infty$ embedding $f:S^2\to{\mathbb{R}}^3$ is a smooth injective immersion that is a homeomorphism onto its image. Six points form the vertices of a nondegenerate regular octahedron if they are precisely \[ \{c+r u_1,c-r u_1,c+r u_2,c-r u_2,c+r u_3,c-r u_3\}, \] where $c\in{\mathbb{R}}^3$, $r>0$, and $(u_1,u_2,u_3)$ is an orthonormal basis of ${\mathbb{R}}^3$.

Question. For every $C^\infty$ embedding $f:S^2\to{\mathbb{R}}^3$, do there exist $c\in{\mathbb{R}}^3$, $r>0$, and an orthonormal basis $(u_1,u_2,u_3)$ such that \[ c+r u_i\in f(S^2)\quad\text{and}\quad c-r u_i\in f(S^2) \qquad\text{for every }i\in\{1,2,3\}? \] Only the six vertices are required to lie on the embedded sphere; containment of the edges, faces, or solid octahedron in the surface is not required. The supplied background also mentions metric two-spheres, but gives no definition or hypotheses for that broader formulation; the question here retains the explicitly stated smooth embedded setting.

Short English statement

Does every smoothly embedded two-sphere in ${\mathbb{R}}^3$ contain the six vertices of a nondegenerate regular octahedron? The supplied record labels this problem open; that status is supplied curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30001142$, title: Regular Octahedra Inscribed in Smooth Two-Spheres; source problem code catalogue entry. The record attributes the problem to the Oberwolfach Reports collection and gives proposed year 2009, without naming an individual proposer.
• Discrete Differential Geometry (2009), Oberwolfach Reports, pp. 75--144, DOI 10.4171/owr/2009/02. Supplied source report and literature-source link: https://doi.org/10.4171/owr/2009/02. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Matschke, square peg problem and relatives, as labeled in the supplied literature sources: https://doi.org/10.1090/tran/8649. The supplied assessment says that the regular-octahedron problem for smooth or metric two-spheres remains open and that Matschke identifies the dimension-three crosspolytope case as open after finding a topological obstruction. This assessment is supplied dated curation checked on 2026-08-21, not independently verified here; its status and cited claims require independent verification.
• The registry marks the statement corrected_verified and describes its processing as meaning-preserving semantic, typographic, and LaTeX cleanup without unsupported source-dependent additions. These are supplied provenance labels, not an independent mathematical or source verification of this draft.