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.

翻译：Kalai的$3^d$猜想指出，每个中心对称的$d$维多面体至少拥有$3^d$个面。我们给出两个特殊情形的简短证明：当$P$是无条件的（即关于任意坐标超平面的反射不变），以及更一般地，当$P$是局部反阻的（即在每个卦限中表现为一个无条件多面体）。在这两种情形中，我们证明最小值恰好由Hanner多面体达到。