Hanner polytope

[2] Alternatively and equivalently to the polar dual operation, the Hanner polytopes may be constructed by Cartesian products and direct sums, the dual of the Cartesian products.

This direct sum operation combines two polytopes by placing them in two linearly independent subspaces of a larger space and then constructing the convex hull of their union.

[3][4] A cube is a Hanner polytope, and can be constructed as a Cartesian product of three line segments.

Its dual, the octahedron, is also a Hanner polytope, the direct sum of three line segments.

[8] The translates of a hypercube (or of an affine transformation of it, a parallelotope) form a Helly family: every set of translates that have nonempty pairwise intersections has a nonempty intersection.

[9] For any other centrally symmetric convex polytope K, Hanner (1956) defined I(K) to be the smallest number of translates of K that do not form a Helly family (they intersect pairwise but have an empty intersection).

[4] For d = 1, 2, 3, ... it is: A more explicit bijection between the Hanner polytopes of dimension d and the cographs with d vertices is given by Reisner (1991).

For this bijection, the Hanner polytopes are assumed to be represented geometrically using coordinates in {0,1,−1} rather than as combinatorial equivalence classes; in particular, there are two different geometric forms of a Hanner polytope even in two dimensions, the square with vertex coordinates (±1,±1) and the diamond with vertex coordinates (0,±1) and (±1,0).

Conversely, every cograph can be represented in this way by a Hanner polytope.