Minimum entropy of a log-concave variable with fixed variance

• 成果类型: Article
• 署名作者: Melbourne, James; Nayar, Piotr; Roberto, Cyril
• 署名单位: CIMAT - Centro de Investigacion en Matematicas; University of Warsaw; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-025-01431-3
• 发表日期: 2026-04
• 页码: 2173-2191
• 关键词: entropy; variance; log-concave random variables; POWER INEQUALITY

摘要

We show that for log-concave real random variables with fixed variance the Shannon differential entropy is minimized for an exponential random variable, answering a 2010 question of Bobkov and Madiman [1]. This gives a sharp reversal of the celebrated entropy maximization theorem due to Boltzmann, in the log-concave case. We apply this result to derive upper bounds on capacities of additive noise channels with log-concave noise. We also improve constants in the reverse entropy power inequalities for log-concave random variables.