Logarithmically concave measure

In mathematics, A Borel measure "μ" on "n"-dimensional Euclidean space 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

: $mu(lambda\; A\; +\; (1-lambda)\; B)\; geq\; mu(A)^lambda\; mu(B)^\{1-lambda\},$

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

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 paper | author=Borell, C. | title=Convex set functions in d-space | date = 1975 ] , 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.

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