Kalai's $3^{d}$-conjecture for unconditional and locally anti-blocking polytopes

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