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

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