Sophomore's dream: Difference between revisions
en>PMajer →Proof: (alternatively, we may keep "converting the integral" but then I'd say: "to the Euler integral etc", as written below) |
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In [[mathematics]], a [[Borel measure]] ''μ'' on ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup> is called '''logarithmically concave''' (or '''log-concave''' for short) if, for any [[compact set|compact subsets]] ''A'' and ''B'' of '''R'''<sup>''n''</sup> and 0 < ''λ'' < 1, one has | |||
: <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.|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== | |||
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. | |||
By a theorem of Borell,<ref>{{cite paper | 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}}</ref> 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== | |||
<references/> | |||
[[Category:Measures (measure theory)]] | |||
Revision as of 05:12, 29 January 2014
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
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Template:Cite paper