Logarithmically concave measure explained

R ^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of R ^{n} and 0 < λ < 1, one has\mu( λ A + ( 1-λ ) B) \geq \mu(A) ^{λ} \mu(B) ^{1-λ} ,

where λ A + (1 − λ) B denotes the Minkowski sum of λ A and (1 − λ) B.^[1]

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

Notes and References

1. Book: Prékopa, A.. 0592596. András Prékopa. Logarithmic concave measures and related topics. Stochastic programming (Proc. Internat. Conf., Univ. Oxford, Oxford, 1974). 63–82. Academic Press. London-New York. 1980.
2. Borell, C. . Convex set functions in d-space . 1975 . 0404559. Period. Math. Hungar. . 6. 2. 111–136. 10.1007/BF02018814. 122121141 .