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In [[mathematics]], a [[Borel measure]] ''&mu;'' on ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup> is called '''logarithmically concave''' (or '''log-concave''' for short) if, for any [[compact set|compact subsets]] ''A'' and ''B'' of '''R'''<sup>''n''</sup> and 0&nbsp;&lt;&nbsp;''&lambda;''&nbsp;&lt;&nbsp;1, one has

: <math> \mu(\lambda A + (1-\lambda) B) \geq \mu(A)^\lambda \mu(B)^{1-\lambda}, </math>

where ''&lambda;''&nbsp;''A''&nbsp;+&nbsp;(1&nbsp;&minus;&nbsp;''&lambda;'')&nbsp;''B'' denotes the [[Minkowski sum]] of ''&lambda;''&nbsp;''A'' and (1&nbsp;&minus;&nbsp;''&lambda;'')&nbsp;''B''.<ref>{{cite book|mr=0592596|last=Prékopa|first=A.|author-link=András Prékopa|chapter=Logarithmic concave measures and related topics|title=Stochastic programming (Proc. Internat. Conf., Univ. Oxford, Oxford, 1974)|pages=63&ndash;82|publisher=Academic Press|location=London-New York|year=1980}}</ref>

==Examples==

The [[Brunn-Minkowski theorem|Brunn-Minkowski inequality]] asserts that the [[Lebesgue measure]] is log-concave. The restriction of the Lebesgue measure to any [[convex set]] is also log-concave. 

By a theorem of Borell,<ref>{{cite paper | author=Borell, C. | title=Convex set functions in ''d''-space | year = 1975 | mr=0404559|journal=Period. Math. Hungar. |volume=6|issue=2|pages=111&ndash;136|doi=10.1007/BF02018814}}</ref> a measure is log-concave if and only if it has a density with respect to the Lebesgue measure on some affine hyperplane, and this density is a [[logarithmically concave function]]. Thus, any [[Gaussian measure]] is log-concave.

The [[Prékopa–Leindler inequality]] shows that a [[convolution]] of log-concave measures is log-concave.

==References==

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[[Category:Measures (measure theory)]]