Ziegler’s maximal-vertex binary-polytope characterization
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Ziegler’s bound limits a simplicial d-dimensional 0/1-polytope to 2d vertices. Must every polytope attaining that count be centrally symmetric and hence fit the proposed cross-polytope characterization?
Why it matters and what counts as progress
Why it matters: An extremal vertex count need not force the combinatorial or symmetry structure suggested by low-dimensional examples.
What counts as progress: A characterization proof or an explicit simplicial binary polytope with 2d vertices lacking central symmetry, with dimension and simpliciality certified.
How the problem developed
- 30 Jun 2026 · Counterexample [1]
Volker Kaibel and Sebastian Pokutta Exhibited 14 vertices in {0,1}⁷ whose convex hull is a simplicial 7-polytope but is not centrally symmetric, disproving the maximal-vertex characterization.
AI contribution
A Counterexample to Ziegler's Cross-Polytope Conjecture for Simplicial 0/1-Polytopes
Fourteen binary vertices form a simplicial seven-dimensional 0/1-polytope with the maximum 2d vertices but without central symmetry, countering the proposed cross-polytope characterization.
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Publication dates and editorial changes.
Agent-guided construction refutes Ziegler cross-polytope conjecture