A strong log-concavity property for measures on Boolean algebras

TL;DR

This paper introduces the antipodal pairs property for measures on Boolean algebras, showing it implies strong log-concavity and applying it to improve existing results, prove preservation of ultra-log-concavity, and advance a conjecture.

Contribution

It establishes a new antipodal pairs property that links to log-concavity, providing novel proofs and progress on longstanding conjectures in combinatorics.

Findings

Antipodal pairs property implies strong log-concavity.

Improves results of Wagner on log-concavity.

Provides a new proof of Liggett's theorem on ultra-log-concavity.

Abstract

We introduce the antipodal pairs property for probability measures on finite Boolean algebras and prove that conditional versions imply strong forms of log-concavity. We give several applications of this fact, including improvements of some results of Wagner; a new proof of a theorem of Liggett stating that ultra-log-concavity of sequences is preserved by convolutions; and some progress on a well-known log-concavity conjecture of J. Mason.