{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T01:04:05Z","timestamp":1779239045813,"version":"3.51.4"},"reference-count":38,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,3,9]],"date-time":"2018-03-09T00:00:00Z","timestamp":1520553600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Walter S. Baer and Jeri Weiss CMI Postdoctoral Fellowship"},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1566567"],"award-info":[{"award-number":["CCF-1566567"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We derive a lower bound on the differential entropy of a log-concave random variable X in terms of the p-th absolute moment of X. The new bound leads to a reverse entropy power inequality with an explicit constant, and to new bounds on the rate-distortion function and the channel capacity. Specifically, we study the rate-distortion function for log-concave sources and distortion measure \r\n          \r\n            \r\n              \r\n                d\r\n                \r\n                  (\r\n                  x\r\n                  ,\r\n                  \r\n                    x\r\n                    ^\r\n                  \r\n                  )\r\n                \r\n                =\r\n                \r\n                  \r\n                    |\r\n                    x\r\n                    \u2212\r\n                    \r\n                      x\r\n                      ^\r\n                    \r\n                    |\r\n                  \r\n                  r\r\n                \r\n              \r\n            \r\n          \r\n        , with \r\n          \r\n            \r\n              \r\n                r\r\n                \u2265\r\n                1\r\n              \r\n            \r\n          \r\n        , and we establish that the difference between the rate-distortion function and the Shannon lower bound is at most \r\n          \r\n            \r\n              \r\n                log\r\n                (\r\n                \r\n                  \r\n                    \u03c0\r\n                    e\r\n                  \r\n                \r\n                )\r\n                \u2248\r\n                1\r\n                .\r\n                5\r\n              \r\n            \r\n          \r\n         bits, independently of r and the target distortion d. For mean-square error distortion, the difference is at most \r\n          \r\n            \r\n              \r\n                log\r\n                (\r\n                \r\n                  \r\n                    \r\n                      \u03c0\r\n                      e\r\n                    \r\n                    2\r\n                  \r\n                \r\n                )\r\n                \u2248\r\n                1\r\n              \r\n            \r\n          \r\n         bit, regardless of d. We also provide bounds on the capacity of memoryless additive noise channels when the noise is log-concave. We show that the difference between the capacity of such channels and the capacity of the Gaussian channel with the same noise power is at most \r\n          \r\n            \r\n              \r\n                log\r\n                (\r\n                \r\n                  \r\n                    \r\n                      \u03c0\r\n                      e\r\n                    \r\n                    2\r\n                  \r\n                \r\n                )\r\n                \u2248\r\n                1\r\n              \r\n            \r\n          \r\n         bit. Our results generalize to the case of a random vector X with possibly dependent coordinates. Our proof technique leverages tools from convex geometry.<\/jats:p>","DOI":"10.3390\/e20030185","type":"journal-article","created":{"date-parts":[[2018,3,9]],"date-time":"2018-03-09T12:17:41Z","timestamp":1520597861000},"page":"185","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":30,"title":["A Lower Bound on the Differential Entropy of Log-Concave Random Vectors with Applications"],"prefix":"10.3390","volume":"20","author":[{"given":"Arnaud","family":"Marsiglietti","sequence":"first","affiliation":[{"name":"Center for the Mathematics of Information, California Institute of Technology, Pasadena, CA 91125, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Victoria","family":"Kostina","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1109\/18.119699","article-title":"On Universal Quantization by Randomized Uniform\/Lattice Quantizers","volume":"32","author":"Zamir","year":"1992","journal-title":"IEEE Trans. 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