A note on Kalai's $3^d$ Conjecture

TL;DR

This paper proves Kalai's conjecture that a centrally symmetric convex polytope in d dimensions has at least 3^d facets, under the condition of symmetry about d orthogonal hyperplanes.

Contribution

The paper establishes the conjecture for polytopes symmetric about d orthogonal hyperplanes, providing a partial proof of Kalai's 1989 conjecture.

Findings

Kalai's conjecture holds for polytopes symmetric about d orthogonal hyperplanes.

The result confirms the minimum number of facets in this symmetric case.

Provides insight into the structure of centrally symmetric convex polytopes.

Abstract

Suppose that $C$ is a centrally symmetric $d$-dimensional convex polytope; in 1989 Kalai conjectured that $C$ has at least $3^d$ facets. We prove this result if there are $d$ hyperplanes with orthogonal normal vectors so that $C$ is symmetric about all of them.