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

TL;DR

This paper proves special cases of Kalai's $3^d$ conjecture, showing that unconditional and locally anti-blocking polytopes attain the minimal face count with Hanner polytopes.

Contribution

It provides concise proofs for the conjecture in specific classes of polytopes, expanding understanding of face counts in symmetric polytopes.

Findings

Unconditional polytopes satisfy Kalai's conjecture with Hanner polytopes as minimizers.

Locally anti-blocking polytopes also attain the minimum face count with Hanner polytopes.

The proofs are short and focus on special symmetric classes of polytopes.

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.