2026/06/30 by Volker Kaibel, Sebastian Pokutta · 1 voice
Mathematics · #math.CO #math.OC
arxiv published 2026/06/30 · arxiv updated 2026/07/02
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.