Logarithmically concave measure

In mathematics, a Borel measure μ on n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of R n {\displaystyle \mathbb {R} ^{n}} and 0 < λ < 1, one has μ ( λ A + ( 1 − λ ) B ) ≥ μ ( A ) λ μ ( B ) 1 − λ , {\displaystyle \mu (\lambda A+(1-\lambda )B)\geq \mu (A)^{\lambda }\mu (B)^{1-\lambda },} 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.

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