Logarithmically concave measure
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 and 0 < λ < 1, one has
:
where λ A + (1 − λ) B denotes the Minkowski sum of λ A and (1 − λ) B.{{cite book|mr=0592596|last=Prékopa|first=A.|author-link=András Prékopa|chapter=Logarithmic concave measures and related topics|title=Stochastic programming (Proc. Internat. Conf., Univ. Oxford, Oxford, 1974)|pages=63–82|publisher=Academic Press|location=London-New York|year=1980}}
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,{{cite journal | author=Borell, C. | title=Convex set functions in d-space | year = 1975 | mr=0404559|journal=Period. Math. Hungar. |volume=6|issue=2|pages=111–136|doi=10.1007/BF02018814| s2cid=122121141 }} 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