Logarithmically concave measure: Difference between revisions
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In [[mathematics]], |
In [[mathematics]], a [[Borel measure]] ''μ'' on ''n''-[[dimension]]al [[Euclidean space]] <math>\mathbb{R}^{n}</math> is called '''logarithmically concave''' (or '''log-concave''' for short) if, for any [[compact set|compact subsets]] ''A'' and ''B'' of <math>\mathbb{R}^{n}</math> and 0 < ''λ'' < 1, one has |
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: <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> |
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where '' |
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> |
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==Examples== |
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The [[Brunn-Minkowski theorem|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. |
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The [[Brunn–Minkowski theorem|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. |
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By a theorem of Borell,<ref>{{cite journal | author=Borell, C. | title=Convex set functions in ''d''-space | year = 1975 | mr=0404559|journal=Period. Math. Hungar. |volume=6|issue=2|pages=111–136|doi=10.1007/BF02018814| s2cid=122121141 }}</ref> 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. |
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The [[Prékopa–Leindler inequality]] shows that a [[convolution]] of log-concave measures is log-concave. |
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==See also== |
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* [[Convex measure]], a generalisation of this concept |
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* [[Logarithmically concave function]] |
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==References== |
==References== |
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{{reflist}} |
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<references/> |
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{{Measure theory}} |
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[[hu:Logkonkáv mérték]] |
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Latest revision as of 01:47, 15 January 2023
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
[edit]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
[edit]- Convex measure, a generalisation of this concept
- Logarithmically concave function
References
[edit]- ^ 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.