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Logarithmically concave measure
In mathematics, a Borel measure ''μ'' on ''n''-dimensional Euclidean space R''n'' is called logarithmically concave (or log-concave for short) if, for any compact subsets ''A'' and ''B'' of R''n'' and 0 < ''λ'' < 1, one has
:
where ''λ'' ''A'' + (1 − ''λ'') ''B'' denotes the Minkowski sum of ''λ'' ''A'' and (1 − ''λ'') ''B''.
==Examples==
The 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, 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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