Minimum Maximum Degree of Centrally Symmetric Sphere Triangulations

Definitions and mathematical statement

For each integer $d\geq 1$, consider the following precise candidate interpretation of the source question. Write \[ S^d=\{x\in{\mathbb{R}}^{d+1}:\|x\|=1\}. \] Let $K,\tau)$ consist of a finite simplicial complex $K$ and a simplicial involution $\tau$, with a homeomorphism $h:|K|\to S^d$ satisfying \[ h\circ|\tau|=-h. \] Thus the symmetry corresponds to the antipodal map on the sphere. For a vertex $v$, define its degree and the maximum vertex degree by \[ \deg_K(v)=\#\{w\in V(K)\setminus\{v\}:\{v,w\}\in K\}, \qquad \Delta(K)=\max_{v\in V(K)}\deg_K(v). \] Under this interpretation, set \[ f_{\mathrm{sph}}(d)=\min_{(K,\tau)}\Delta(K). \] For the polytope formulation, let $\mathcal P_d$ be the class of full-dimensional convex simplicial polytopes $P\subset{\mathbb{R}}^{d+1}$ with $P=-P$; translating the center to the origin does not change vertex degrees. Define \[ f_{\mathrm{poly}}(d)=\min_{P\in\mathcal P_d}\Delta(\partial P), \] where $\partial P$ denotes the boundary simplicial complex. These classes are nonempty, as witnessed by the boundary of the $(d+1)$-dimensional cross-polytope. Every such polytope boundary gives an antipodally symmetric sphere triangulation, so \[ f_{\mathrm{sph}}(d)\leq f_{\mathrm{poly}}(d). \] Requested determination and unresolved scope. Determine the minimum possible maximum vertex degree as a function of $d$. The supplied record calls this function $f(d)$ and describes the sphere and polytope formulations as equivalent. It does not specify whether sphere triangulations are restricted to polytopal ones, nor does it supply a justification that the two minima agree. Consequently, the definitions above are explicit candidate formulations, and their equality is not asserted here as a verified fact. The intended meaning of central symmetry and the intended class of triangulations require confirmation against the source.

The record reports \[ f(1)=2,\qquad f(2)=4,\qquad d+1<f(d)\leq 2d. \] These are supplied claims, not independently verified results in this draft. The strict lower bound contradicts the reported value at $d=1$. Its intended dimension range is missing; no corrected range is silently imposed. Positive dimensions are used for the candidate formulations above because the record explicitly discusses $d=1,2$; it gives no convention for $d=0$.

Short English statement

Determine the smallest achievable maximum vertex degree among centrally symmetric triangulations of a sphere. The source also gives a simplicial polytope formulation, but the claimed equivalence needs clarification. It reports values in dimensions $1$ and $2$ and general bounds whose stated range conflicts with the dimension-$1$ value. The supplied dated curation labels the problem open; this is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001134$, source problem code catalogue entry, entitled Minimum Maximum Degree of Centrally Symmetric Sphere Triangulations, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2009$ and names no individual proposer.
• Supplied report citation: Discrete Differential Geometry (2009), Oberwolfach Reports, pp. 75--144, DOI 10.4171/owr/2009/02. https://doi.org/10.4171/owr/2009/02. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; no problem-specific page is supplied.
• Supplied literature source: Rote, Open Problems in Discrete Differential Geometry. https://page.mi.fu-berlin.de/rote/Kram/OWR-DDG09-problems.pdf.
• Supplied literature source labeled "original OWR report": https://doi.org/10.4171/owr/2009/02.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, says that the exact function remains undetermined in the checked literature, and reports $d+1<f(d)\leq2d$. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.