title: Log-Concavity and Strong Log-Concavity: a review.
abstract: 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 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.
authors.affiliation: Departamento de Estadística, CIMFAV, Universidad de Valparaíso, Chile.
authors.forename: Adrien
authors.initials: A
authors.lastname: Saumard
authors.affiliation: Department of Statistics, Box 354322, University of Washington, Seattle, WA 98195-4322.
authors.forename: Jon A
authors.initials: JA
authors.lastname: Wellner
keywords: concave; convex; convolution; inequalities; log-concave; monotone; preservation; strong log-concave