Neighborliness of Centrally Symmetric Polytopes

Definitions and mathematical statement

Extremal neighborliness problem. How neighborly can a centrally symmetric polytope be as a function of its dimension and number of vertices? For a full-dimensional $d$-polytope, translate its symmetry center to zero, so $P=-P$. Its vertices occur in antipodal pairs $v,-v$; hence the vertex count $N$ is even, with the usual admissible range $N\ge2d$.

In the central-symmetry-adapted convention, $P$ is $k$-neighborly if every subset of at most $k$ vertices containing no antipodal pair is the vertex set of a proper face. A precise extremal target is\[\nu(d,N)=\max\{k\in\{0,\ldots,d\}:\exists\text{ centrally symmetric }d\text{-polytope with }N\text{ vertices having this property}\}.\] Determine this function, or sharp asymptotic bounds in stated regimes of $d,N$, together with constructions achieving the proposed order. The explicit range $k\le d$ avoids a vacuous unbounded parameter when only a limited number of nonantipodal vertices can be selected. No asymptotic candidate or required approximation factor is supplied.

The source does not define neighborliness. Excluding antipodal pairs is essential to the intended centrally symmetric version: an antipodal segment passes through the interior and cannot be an edge, so ordinary two-neighborliness would already fail. The adapted convention above is therefore explicit and requires source confirmation, not silently confused with ordinary neighborliness of unrestricted polytopes. The source asks for dependence on both dimension and vertex count, not merely examples with a fixed number of vertices.

The supplied triage checked 2026-08-21 labels the sharp asymptotic problem open, reporting substantial constructions and upper bounds without a sharp general answer. It supplies no exact quantitative theorem in the record. This dated assessment requires independent verification.

Short English statement

Determine the maximum central-symmetry-adapted neighborliness for each admissible dimension and vertex count, or its sharp asymptotic order. Nonantipodal vertex subsets are the relevant ones; ordinary neighborliness would be blocked immediately by antipodal pairs.

Sources

• Local UnsolvedMath registry, permanent ID 30000691, code catalogue entry; Neighborliness of Centrally Symmetric Polytopes, Oberwolfach Reports Open Problems.
• Geometric and Topological Combinatorics (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/04, pp. 195--272. https://doi.org/10.4171/owr/2007/04. Proposed year 2007; no proposer named.
• Supplied literature label original problem report: https://doi.org/10.4171/owr/2007/04.
• Supplied literature label Novik--Ziegler constructions: https://arxiv.org/abs/math/0507280.
• Supplied literature label explicit constructions and bounds: https://arxiv.org/abs/1203.6867.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. Statement status exact, reported retained byte-for-byte. Verify dated claims independently.