Kalai's 3^d conjecture
In geometry, Kalai's 3 conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989. It states that every "d"-dimensional centrally symmetric polytope has at least 3 nonempty faces (including the polytope itself as a face but not including the empty set). Examples. In two dimensions, the simplest centrally symmetric convex polygons are the parallelograms, which have four vertices, four edges, and one polygon: . A cube is centrally symmetric, and has 8 vertices, 12 edges, 6 square sides, and 1 solid: . Another three-dimensional convex polyhedron, the regular  octahedron, is also centrally symmetric, and has 6 vertices, 12 edges, 8 triangular sides, and 1 solid: . In higher dimensions, the hypercube [0, 1] has exactly 3 faces, each of which can be determined by specifying, for each of the "d" coordinate axes, whether the face projects onto that axis onto the point 0, the point 1, or the interval [0, 1]. More generally, every Hanner polytope has exactly 3 faces. If Kalai's conjecture is true, these polytopes would be among the centrally symmetric polytopes with the fewest possible faces. Generalizations. In the same work as the one in  which the 3 conjecture appears, Kalai conjectured more strongly which the "f"-vector of every convex centrally symmetric polytope "P" dominates the "f"-vector of at least one Hanner polytope "H" of the same dimension. This means that, for every number "i" from 0 to the dimension of "P", the number of "i"-dimensional faces of "P" is greater than or equal to the number of "i"-dimensional faces of "H". If it were true, this would imply the truth of the 3 conjecture; however, the stronger conjecture was later disproven. Status. The conjecture is known to be true for formula_1. It is also  known to be true for simplicial polytopes: it follows in this case from a conjecture of that every centrally symmetric simplicial polytope has at least as many faces of each dimension as the cross polytope, proven by . Indeed, these two previous papers were cited by Kalai as part of the basis for making his conjecture. Another special class of polytopes that the conjecture has been proven for are the Hansen polytopes of split graphs, which had been used by to disprove the stronger conjectures of Kalai. The 3 conjecture remains open for arbitrary polytopes in higher dimensions. Kalai's 3^d