A surprisingly interesting family of polytopes Hd defined by the inequalites -1 <= xi <= 1. Finding the largest volume simplex in d-cube is an ongoing project [128, 127] as is finding the smallest possible triangulation . A skewed class of cubes gave the first hard examples for the simplex method . Cubes are conjectured to maximize diameter for d-polytopes with 2d facets [174, 166, 165]. For any triangle-free polytope P, fk(P) >= fk(Hd) ; this same inequality is conjectured to hold for centrally-symmetric polytopes . Finally, cubes are conjectured to maximize the number of pairs of estranged vertices (i.e. not sharing a facet) for simple d polytopes with 2d facets .
vertices>>facets, simple, triangle-free, facet-degenerate, centrally-symmetric, triangulation, zero-one, zonotope