Logarithmically concave measure

From Wikipedia, the free encyclopedia

A Borel measure μ on the Euclidean space Rn is called logarithmically concave (or log-concave for short) if for any compact sets A and B in Rn and 0 < λ < 1 one has

\mu(\lambda A + (1-\lambda) B) \geq \mu(A)^\lambda \mu(B)^{1-\lambda}

where λA + (1 − λ)B denotes the Minkowski sum of λA and (1 − λ)B.

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[1], a measure 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, the Gaussian measure is log-concave.

[edit] References

  1. ^ Borell, C. (1975). "Convex set functions in d-space".
Retrieved from "http://en.wikipedia.org../../../l/o/g/Logarithmically_concave_measure.html"