In mathematics, a Borel measure ü on n-dimensional Euclidean space is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of and 0 < û < 1, one has
where û A + (1 â û) B denotes the Minkowski sum of û A and (1 â û) B.
The BrunnâÂÂMinkowski 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 PrékopaâÂÂLeindler inequality shows that a convolution of log-concave measures is log-concave.