  We define an analogue of the cube and an analogue of the 5-wedge in higher
dimensions, each with $2d+2$ vertices and $d^2+2d-3$ edges. We show that these
two are the only minimisers of the number of edges, amongst d-polytopes with
$2d+2$ vertices, for all $d$ except 4, 5 and 7. We also show that there are
four sporadic minimisers in these low dimensions. We announce a partial
solution to the corresponding problem for polytopes with $2d + 3$ vertices.
