Kalai’s 3^𝑑-conjecture for unconditional and locally anti-blocking polytopes

Kalai’s 3 d 3ᵈ conjecture states that every centrally-symmetric d d -polytope has at least 3 d 3ᵈ faces. We give short proofs for two special cases: if P P is unconditional (that is, invariant w. r. t. reflection in any coordinate hyperplane), and more generally, if P P is locally anti-blocking. In both cases we show that the minimum is attained exactly for the Hanner polytopes.