{"ID":5438885,"CreatedAt":"2026-07-01T01:17:58.482524686Z","UpdatedAt":"2026-07-03T13:17:43.497842103Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.31640","arxiv_id":"2606.31640","title":"A Counterexample to Ziegler's Cross-Polytope Conjecture for Simplicial 0/1-Polytopes","abstract":"Ziegler proved that every simplicial $d$-dimensional $0/1$-polytope has at most $2d$ vertices, and asked whether equality forces the polytope to be centrally symmetric and hence, equivalently, a $0/1$-realization of the $d$-dimensional cross polytope. In this note, we give a negative answer, exhibiting an explicit set of $14$ vertices in $\\{0,1\\}^7$ whose convex hull is a simplicial $7$-polytope and is not centrally symmetric. Moreover, via exhaustive enumeration we show that up to the symmetries of the cube, there are precisely five such polytopes in dimension $7$ (of two combinatorial types) that are not centrally symmetric.","short_abstract":"Ziegler proved that every simplicial $d$-dimensional $0/1$-polytope has at most $2d$ vertices, and asked whether equality forces the polytope to be centrally symmetric and hence, equivalently, a $0/1$-realization of the $d$-dimensional cross polytope. In this note, we give a negative answer, exhibiting an explicit set...","url_abs":"https://arxiv.org/abs/2606.31640","url_pdf":"https://arxiv.org/pdf/2606.31640v1","authors":"[\"Volker Kaibel\",\"Sebastian Pokutta\"]","published":"2026-06-30T13:21:48Z","proceeding":"math.CO","tasks":"[\"math.CO\",\"math.OC\"]","methods":"[]","has_code":false}