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arXiv · 2308.02909

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

Raman Sanyal·Martin Winter
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Abstract

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.

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Raman Sanyal, Martin Winter. 2024-04-21. Kalai's $3^{d}$ conjecture for unconditional and locally anti-blocking polytopes. https://arxiv.org/abs/2308.02909

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