In mathematics, a Borel measure μ on n-dimensional Euclidean space
is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of
[1] 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,[2] 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.