In mathematics, a Borel measure Ό 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 < λ < 1, one has
where λ A + (1 â λ) B denotes the Minkowski sum of λ A and (1 â λ) B.[1]
Examples
The BrunnâMinkowski 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 PrĂ©kopaâLeindler 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
- ^ PrĂ©kopa, A. (1980). "Logarithmic concave measures and related topics". Stochastic programming (Proc. Internat. Conf., Univ. Oxford, Oxford, 1974). London-New York: Academic Press. pp. 63â82. MR 0592596.
- ^ Borell, C. (1975). "Convex set functions in d-space". Period. Math. Hungar. 6 (2): 111â136. doi:10.1007/BF02018814. MR 0404559. S2CID 122121141.
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