Triangle-free polytopes with few facets (Q811664)

A convex \(d\)-polytope is called triangle-free if it has no triangular 2- face. In this paper, the triangle-free \(d\)-polytopes with up to \(2d+2\) facets are completely enumerated: In higher dimensions there occur seven combinatorial types, and they are all iterated prisms. Furthermore, it is shown that a convex polytope with not too many facets has either a triangular 2-face or a \(k\)-face \((k\geq 2)\) which is combinatorially equivalent to the \(k\)-cube. For given \(k\) an appropriate bound for the number of facets is \(2d+(d-k)\), which cannot be improved for \(k\geq d/2+1\).