Weak Knaster Conjecture

Definitions and mathematical statement

For a positive integer $n$, let \[ S^{n-1}=\{x\in{\mathbb{R}}^n:\|x\|_2=1\},\qquad O(n)=\{\rho\in{\mathbb{R}}^{n\times n}:\rho^{\mathsf T}\rho=I_n\}. \] Here $O(n)$ includes both orientation-preserving and orientation-reversing orthogonal transformations; the source uses the word "rotation" for an element of this group.

Interpreting the point count $\ell$ as a positive integer, the supplied question asks whether \[ \begin{gathered} \forall\ell\in{\mathbb{Z}}_{\geq1}\ \exists n\in{\mathbb{Z}}_{\geq1}\ \forall x_1,\ldots,x_\ell\in S^{n-1}\text{ pairwise distinct}\ \forall f\in C(S^{n-1},{\mathbb{R}}),\\ \exists\rho\in O(n)\quad f(\rho x_1)=\cdots=f(\rho x_\ell). \end{gathered} \] The dimension $n$ may depend on $\ell$, but must be chosen independently of the point set and the continuous function. The transformation $\rho$ may depend on both.

Unresolved qualification. The source supplies no lower bound on $n$ beyond that needed to interpret the sphere. Since $S^0$ has only two points, choosing $n=1$ makes the universal assertion over $\ell$-element subsets empty for $\ell\geq3$. Thus the displayed literal formulation has a vacuity that conflicts with the nontrivial general finiteness question described in the supplied background. A restriction ensuring that such point sets exist, or another intended dimension restriction, requires source confirmation; none is silently added here.

Short English statement

For each positive point count $\ell$, does a dimension exist in which every $\ell$-point subset of the unit sphere can be moved by an orthogonal transformation into one level set of every continuous real-valued function on that sphere? The supplied formulation needs clarification of its dimension restriction to exclude vacuous choices.

Sources

• Local UnsolvedMath registry: supplied record from data/unsolvedmath/problems.json, permanent ID $30001697$, title "Weak Knaster Conjecture," exact source problem code catalogue entry. The record attributes the problem to the 2011 Oberwolfach workshop collection and identifies no individual proposer.
• Supplied source citation: Topological and Geometric Combinatorics (2011), Oberwolfach Reports, DOI 10.4171/owr/2011/08, pp. 349--423. Original OWR report: https://doi.org/10.4171/owr/2011/08.
• Supplied literature reference, labeled "Bukh--Karasev weak Knaster results": https://arxiv.org/abs/1212.5351.
• Literature assessment checked on 2026-08-21: the supplied curation labels the problem partially solved, reports substantial special-case results, and states that the general finiteness assertion remains open and that $n(4)$ was unresolved in the primary literature located. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied editorial claims, not independent mathematical verification of this draft.