  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.
