- Source: Logarithmically concave measure
In mathematics, a Borel measure μ on n-dimensional Euclidean space
R
n
{\displaystyle \mathbb {R} ^{n}}
is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of
R
n
{\displaystyle \mathbb {R} ^{n}}
and 0 < λ < 1, one has
μ
(
λ
A
+
(
1
−
λ
)
B
)
≥
μ
(
A
)
λ
μ
(
B
)
1
−
λ
,
{\displaystyle \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.
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, 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