Logarithmically concave measure
None
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 . [1]
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, [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 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
References
- ↑ Prékopa, A. (1980). "Logarithmic concave measures and related topics". Stochastic programming (Proc. Internat. Conf., Univ. Oxford, Oxford, 1974) . London-New York: Academic Press. pp. 63–82. MR 0592596 .
- ↑ Borell, C. (1975). "Convex set functions in d -space". Period. Math. Hungar . 6 (2): 111–136. doi : 10.1007/BF02018814 . MR 0404559 . S2CID 122121141 .
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