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Kalai's $3^{d}$ conjecture for unconditional and locally anti-blocking polytopes
Published 5 Aug 2023 in math.CO, cs.DM, and math.MG | (2308.02909v3)
Abstract: Kalai's $3d$ conjecture states that every centrally-symmetric $d$-polytope has at least $3d$ 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.
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