Two-neighborly simplicial 0/1-polytopes

Abstract

A 0/1-polytope is the convex hull of a subset of the vertices of the unit cube. Ziegler proved that a simplicial 0/1-polytope of dimension d has at most 2d vertices and asked whether the equality case is always a cross polytope; Kaibel and Pokutta recently answered this in the negative with an explicit example in dimension 7, and asked in turn whether a simplicial 0/1-polytope can be 2-neighborly at all. We answer their question affirmatively by exhibiting simplicial 2-neighborly 0/1-polytopes in dimensions 4,5,6,7, with 6,8,10,12 vertices respectively; the three of dimension at least 5 have, in addition, no cube-antipodal pair of vertices, which is the stronger form in which the question is posed. Each example is certified twice and independently: through a complete facet enumeration, and through an explicit integer separating functional for every pair of vertices. We complement this with complete censuses of every subset of {0,1}³ and of {0,1}⁴, which show that in those dimensions the largest 2-neighborly simplicial example has 2d-2 vertices and that none of the 2d-vertex examples is 2-neighborly, and we recompute the invariants of Kaibel and Pokutta's dimension-7 examples from their definitions rather than from their tables. Every theorem and proposition below is machine-checked in Lean 4.

Open review

This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.

Archived files

1. Version 1 · current (opens in a new tab)

   Source snapshot 2026-08-30 15:34 UTC

   File fingerprint

   a35d9b3a26cb70f737e2959180444413304de734d6ae79e8920e4c6355712b95