A surprisingly interesting family of polytopes

H_{d}defined by the inequalites -1 <=x_{i}<= 1. Finding the largest volume simplex ind-cube is an ongoing project [128, 127] as is finding the smallest possible triangulation [126]. A skewed class of cubes gave the first hard examples for the simplex method [78]. Cubes are conjectured to maximize diameter ford-polytopes with 2dfacets [174, 166, 165]. For any triangle-free polytopeP,f_{k}(P) >=f_{k}(H_{d}) [145]; this same inequality is conjectured to hold for centrally-symmetric polytopes [169]. Finally, cubes are conjectured to maximize the number of pairs of estranged vertices (i.e. not sharing a facet) for simpledpolytopes with 2dfacets [146].

vertices>>facets, simple, triangle-free, facet-degenerate, centrally-symmetric, triangulation, zero-one, zonotope

interval

d

2d

2

^{d}

ksum(H([-1,1]),d)

V([-1,1])

^{d}