Math Conjectures

Kalai's 3d3^d Conjecture

OPENMajorConjectureProposed 1989 · Standard version

Canonical statement

Every centrally symmetric convex dd-polytope has at least 3d3^d nonempty faces in total, counting faces of every dimension 0,…,d0,\ldots,d and counting the polytope itself.
View source LaTeX
Every centrally symmetric convex \(d\)-polytope has at
least \(3^d\) nonempty faces in total, counting faces of every dimension
\(0,\ldots,d\) and counting the polytope itself.
DG-011:[Kalai1989CS][SanyalWernerZiegler2009][AdiprasitoEtAl2023ThreePower]
Discrete geometryGeometry and topology

Notes

Kalai conjectured in 1989 that every centrally symmetric convex dd-dimensional polytope has at least 3d3^d nonempty faces, where the count runs over all dimensions from the vertices up to and including the polytope itself [Kalai1989CS]. The proposed bound is exactly achieved by the dd-cube: a face of the cube is specified by choosing, independently in each coordinate, one of three possibilities, giving 3d3^d faces in all. By polar duality the cross-polytope attains the same count, so the cube is not the only candidate extremal.

Partial results are substantial. Sanyal, Werner and Ziegler carried out a detailed study of Kalai's conjectures concerning centrally symmetric polytopes, establishing the 3d3^d bound in low dimensions [SanyalWernerZiegler2009], and Adiprasito, Sanyal and Winter have since proved the bound for substantial classes of centrally symmetric polytopes [AdiprasitoEtAl2023ThreePower].

Despite this progress, the universal statement — that no centrally symmetric dd-polytope whatsoever beats the cube's count — remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.