In mathematics, a Borel measure μ on n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of R n {\displaystyle \mathbb {R} ^{n}} and 0 < λ < 1, one has
μ ( λ A + ( 1 − λ ) B ) ≥ μ ( A ) λ μ ( B ) 1 − λ , {\displaystyle \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,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. Thus, any Gaussian measure is log-concave.
The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave.
See also
- Convex measure, a generalisation of this concept
- Logarithmically concave function
References
Prékopa, A. (1980). "Logarithmic concave measures and related topics". Stochastic programming (Proc. Internat. Conf., Univ. Oxford, Oxford, 1974). London-New York: Academic Press. pp. 63–82. MR 0592596. /wiki/Andr%C3%A1s_Pr%C3%A9kopa ↩
Borell, C. (1975). "Convex set functions in d-space". Period. Math. Hungar. 6 (2): 111–136. doi:10.1007/BF02018814. MR 0404559. S2CID 122121141. /wiki/Doi_(identifier) ↩