title: Logarithmically concave measure
text: # Logarithmically concave measure
  
  In mathematics, a Borel measure μ on n-dimensional Euclidean space $\mathbb {R} ^{n}$ is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of $\mathbb {R} ^{n}$ and 0 < λ < 1, one has
  
      $\mu (\lambda A+(1-\lambda )B)\geq \mu (A)^{\lambda }\mu (B)^{1-\lambda },$
  
  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.
  
  By a theorem of Borell, a probability measure on R^d 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.
  
  