Kalai's conjecture for coordinate-symmetric polytopes

Our understanding of the combinatorics and geometry of centrally symmetric polytopes is still severely limited. This is best illustrated by two elementary yet wildly open conjectures due to Kalai and Mahler. Kalai's 3^d conjecture asserts that the d-cube has the minimal number of faces among all centrally symmetric d-polytopes (that is, 3^d many). Mahler's conjecture asserts that the d-cube has the minimal Mahler volume (volume times volume of dual) among all centrally symmetric d-polytopes. Both conjectures claim the same family of minimizers (the Hanner polytopes) and generally show many parallels, though their connection is not well understood. A polytope is coordinate-symmetric (or unconditional) if it is invariant under reflection on coordinate hyperplanes. While Mahler's conjecture was spectacularly solved for this class of polytopes already in 1987, the same special case was still open for Kalai's conjecture. In joint work with Raman Sanyal we found two very short proofs for Kalai's conjecture for coordinate-symmetric (actually, locally coordinate-symmetric) polytope, that I will present in this talk.