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. 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$d$维多胞体至少拥有$3^d$个面。我们给出了两种特殊情况下的简短证明：当$P$是无条件的（即关于任何坐标超平面的反射下保持不变）时，以及更一般地，当$P$是局部反阻塞时。在这两种情况下，我们证明了最小值恰好由Hanner多胞体达到。