Minimum Number of Edges of Polytopes with $2d+2$ Vertices

Abstract

We define two $d$-polytopes, both with $2d+2$ vertices and $(d+3)(d-1)$ edges, which reduce to the cube and the 5-wedge in dimension three. We show that they are the only minimisers of the number of edges, amongst all $d$-polytopes with $2d+2$ vertices, when $d=6$ or $d\ge8$. We also characterise the minimising polytopes for $d=4, 5$ or 7, where four sporadic examples arise.