Kalai's 3^d Conjecture

Kalai's conjecture states that every centrally symmetric -dimensional convex polytope has at least nonempty faces, counting the polytope itself. The hypercube and cross polytope attain this bound.

The conjecture is known for dimensions at most 4 and for several important classes, including centrally symmetric simplicial polytopes and Hansen polytopes of split graphs. Chambers and Portnoy (2026) proved it whenever the polytope is symmetric about each of the hyperplanes through the origin perpendicular to the vectors of some orthogonal basis.