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. ==See also==