The \( 3^d \) conjecture states that a centrally symmetric \( d \)-dimensional polytope must have at least \( 3^d \) faces (of all dimensions including the \( d \)-dimensional polytope itself as a face but not including the \( -1 \)-dimensional empty set). For instance, the cube has \( 8 \) vertices, \( 12 \) edges, \( 6 \) squares, and \( 1 \) cube as faces; \( 8 + 12 + 6 + 1 = 27 = 3^3. \)
The Hanner polytopes include the cubes, are closed under Cartesian products and duality, and if the conjecture is true have the fewest possible numbers of faces of all centrally symmetric \( d \)-dimensional polytopes. They also have some other interesting properties. In 3d there are only the cube and the octahedron, but in higher dimensions there are a lot more of them.