Log-concavity and strong log-concavity: a review

TL;DR

This paper reviews key results on log-concavity and strong log-concavity in various mathematical contexts, highlighting their preservation properties and connections to other mathematical fields.

Contribution

It provides a unified review, a new proof of Efron's theorem using recent inequalities, and explores interdisciplinary connections of log-concavity.

Findings

Preservation of log-concavity under convolution follows from Efron's monotonicity.

A new proof of Efron's theorem using the asymmetric Brascamp-Lieb inequality.

Connections established between log-concavity and areas like concentration, inequalities, and algorithms.

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 $\mathbb{R}$ 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.