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

08/05/2023

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by   Raman Sanyal, et al.

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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.