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Log-concavity and strong log-concavity: a review

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arxiv 1404.5886 v1 pith:JTAFYMTU submitted 2014-04-23 math.ST stat.TH

Adrien Saumard , Jon A. Wellner This is my paper · ORCID
classification math.STstat.TH
keywords log-concavityreviewefronalgorithmsalongapproximationsareasasymmetric
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. An Improved Bipartition Cover Bound for the Multispecies Coalescent Model

    math.PR 2026-04 unverdicted novelty 7.0 of 10

    Improved upper bounds on the number of loci required for a bipartition cover under the multispecies coalescent, obtained by worst-case analysis of caterpillar and balanced species trees.

  2. Log-concavity and log-convexity in the theory of the Graham--Knuth--Patashnik recurrences

    math.CO 2026-07 accept novelty 6.0 of 10

    GKP arrays T(n,k;µ) are coefficientwise strongly log-concave and their generating polynomials Pn(x;µ) are coefficientwise strongly log-convex (hence Hankel-TP2) when parameters are indeterminates.

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