Logarithmically concave measure: Difference between revisions
ref |
alink |
||
| Line 3: | Line 3: | ||
: <math> \mu(\lambda A + (1-\lambda) B) \geq \mu(A)^\lambda \mu(B)^{1-\lambda}, </math> |
: <math> \mu(\lambda A + (1-\lambda) B) \geq \mu(A)^\lambda \mu(B)^{1-\lambda}, </math> |
||
where ''λ'' ''A'' + (1 − ''λ'') ''B'' denotes the [[Minkowski sum]] of ''λ'' ''A'' and (1 − ''λ'') ''B''.<ref>{{cite book|mr=0592596|last=Prékopa|first=A|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}}</ref> |
where ''λ'' ''A'' + (1 − ''λ'') ''B'' denotes the [[Minkowski sum]] of ''λ'' ''A'' and (1 − ''λ'') ''B''.<ref>{{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}}</ref> |
||
==Examples== |
==Examples== |
||
Revision as of 05:22, 5 September 2011
In mathematics, a Borel measure μ on n-dimensional Euclidean space Rn is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of Rn 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 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, any Gaussian measure is log-concave.
The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave.
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.