Logarithmically concave measure
In mathematics, a Borel measure μ on n-dimensional Euclidean space is called logarithmically concave if, for any compact subsets A and B of and 0 <; λ < 1, one has
where λ A + B denotes the Minkowski sum of λ A and 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.
