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

Raman Sanyal; Martin Winter

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

Combinatorics 2024-04-23 v3 Discrete Mathematics Metric Geometry

Authors: Raman Sanyal , Martin Winter

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.

Keywords

kalai's 3^d conjecture centrally-symmetric polytope unconditional polytope locally anti-blocking polytope hanner polytope face count

Cite

@article{arxiv.2308.02909,
  title  = {Kalai's $3^{d}$ conjecture for unconditional and locally anti-blocking polytopes},
  author = {Raman Sanyal and Martin Winter},
  journal= {arXiv preprint arXiv:2308.02909},
  year   = {2024}
}

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

Abstract

Keywords

Cite

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