A strong log-concavity property for measures on Boolean algebras

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.