  We review and formulate results concerning log-concavity and
strong-log-concavity in both discrete and continuous settings. We show how
preservation of log-concavity and strongly log-concavity on $\mathbb{R}$ under
convolution follows from a fundamental monotonicity result of Efron (1969). We
provide a new proof of Efron's theorem using the recent asymmetric
Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review
connections between log-concavity and other areas of mathematics and
statistics, including concentration of measure, log-Sobolev inequalities,
convex geometry, MCMC algorithms, Laplace approximations, and machine learning.
