In mathematics, the PrĂ©kopaâLeindler inequality is an integral inequality closely related to the reverse Young's inequality, the BrunnâMinkowski inequality and a number of other important and classical inequalities in analysis. The result is named after the Hungarian mathematicians AndrĂĄs PrĂ©kopa and LĂĄszlĂł Leindler. [1] [2]
Let 0 < λ < 1 and let f, g, h : Rn â [0, +â) be non- negative real-valued measurable functions defined on n-dimensional Euclidean space Rn. Suppose that these functions satisfy
(1) |
for all x and y in Rn. Then
Recall that the essential supremum of a measurable function f : Rn â R is defined by
This notation allows the following essential form of the PrĂ©kopaâLeindler inequality: let 0 < λ < 1 and let f, g â L1(Rn; [0, +â)) be non-negative absolutely integrable functions. Let
Then s is measurable and
The essential supremum form was given by Herm Brascamp and Elliott Lieb. [3] Its use can change the left side of the inequality. For example, a function g that takes the value 1 at exactly one point will not usually yield a zero left side in the "non-essential sup" form but it will always yield a zero left side in the "essential sup" form.
It can be shown that the usual PrĂ©kopaâLeindler inequality implies the BrunnâMinkowski inequality in the following form: if 0 < λ < 1 and A and B are bounded, measurable subsets of Rn such that the Minkowski sum (1 − λ)A + λB is also measurable, then
where ÎŒ denotes n-dimensional Lebesgue measure. Hence, the PrĂ©kopaâLeindler inequality can also be used [4] to prove the BrunnâMinkowski inequality in its more familiar form: if 0 < λ < 1 and A and B are non- empty, bounded, measurable subsets of Rn such that (1 − λ)A + λB is also measurable, then
The PrĂ©kopaâLeindler inequality is useful in the theory of log-concave distributions, as it can be used to show that log-concavity is preserved by marginalization and independent summation of log-concave distributed random variables. Since, if have pdf , and are independent, then is the pdf of , we also have that the convolution of two log-concave functions is log-concave.
Suppose that H(x,y) is a log-concave distribution for (x,y) â Rm Ă Rn, so that by definition we have
(2) |
and let M(y) denote the marginal distribution obtained by integrating over x:
Let y1, y2 â Rn and 0 < λ < 1 be given. Then equation ( 2) satisfies condition ( 1) with h(x) = H(x,(1 − λ)y1 + λy2), f(x) = H(x,y1) and g(x) = H(x,y2), so the PrĂ©kopaâLeindler inequality applies. It can be written in terms of M as
which is the definition of log-concavity for M.
To see how this implies the preservation of log-convexity by independent sums, suppose that X and Y are independent random variables with log-concave distribution. Since the product of two log-concave functions is log-concave, the joint distribution of (X,Y) is also log-concave. Log-concavity is preserved by affine changes of coordinates, so the distribution of (X + Y, X â Y) is log-concave as well. Since the distribution of X+Y is a marginal over the joint distribution of (X + Y, X â Y), we conclude that X + Y has a log-concave distribution.
The PrĂ©kopaâLeindler inequality can be used to prove results about concentration of measure.
Theorem[ citation needed] Let , and set . Let denote the standard Gaussian pdf, and its associated measure. Then .
Proof of concentration of measure
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The proof of this theorem goes by way of the following lemma: Lemma In the notation of the theorem, . This lemma can be proven from PrĂ©kopaâLeindler by taking and . To verify the hypothesis of the inequality, , note that we only need to consider , in which case . This allows us to calculate: Since , the PL-inequality immediately gives the lemma. To conclude the concentration inequality from the lemma, note that on , , so we have . Applying the lemma and rearranging proves the result. |
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link)In mathematics, the PrĂ©kopaâLeindler inequality is an integral inequality closely related to the reverse Young's inequality, the BrunnâMinkowski inequality and a number of other important and classical inequalities in analysis. The result is named after the Hungarian mathematicians AndrĂĄs PrĂ©kopa and LĂĄszlĂł Leindler. [1] [2]
Let 0 < λ < 1 and let f, g, h : Rn â [0, +â) be non- negative real-valued measurable functions defined on n-dimensional Euclidean space Rn. Suppose that these functions satisfy
(1) |
for all x and y in Rn. Then
Recall that the essential supremum of a measurable function f : Rn â R is defined by
This notation allows the following essential form of the PrĂ©kopaâLeindler inequality: let 0 < λ < 1 and let f, g â L1(Rn; [0, +â)) be non-negative absolutely integrable functions. Let
Then s is measurable and
The essential supremum form was given by Herm Brascamp and Elliott Lieb. [3] Its use can change the left side of the inequality. For example, a function g that takes the value 1 at exactly one point will not usually yield a zero left side in the "non-essential sup" form but it will always yield a zero left side in the "essential sup" form.
It can be shown that the usual PrĂ©kopaâLeindler inequality implies the BrunnâMinkowski inequality in the following form: if 0 < λ < 1 and A and B are bounded, measurable subsets of Rn such that the Minkowski sum (1 − λ)A + λB is also measurable, then
where ÎŒ denotes n-dimensional Lebesgue measure. Hence, the PrĂ©kopaâLeindler inequality can also be used [4] to prove the BrunnâMinkowski inequality in its more familiar form: if 0 < λ < 1 and A and B are non- empty, bounded, measurable subsets of Rn such that (1 − λ)A + λB is also measurable, then
The PrĂ©kopaâLeindler inequality is useful in the theory of log-concave distributions, as it can be used to show that log-concavity is preserved by marginalization and independent summation of log-concave distributed random variables. Since, if have pdf , and are independent, then is the pdf of , we also have that the convolution of two log-concave functions is log-concave.
Suppose that H(x,y) is a log-concave distribution for (x,y) â Rm Ă Rn, so that by definition we have
(2) |
and let M(y) denote the marginal distribution obtained by integrating over x:
Let y1, y2 â Rn and 0 < λ < 1 be given. Then equation ( 2) satisfies condition ( 1) with h(x) = H(x,(1 − λ)y1 + λy2), f(x) = H(x,y1) and g(x) = H(x,y2), so the PrĂ©kopaâLeindler inequality applies. It can be written in terms of M as
which is the definition of log-concavity for M.
To see how this implies the preservation of log-convexity by independent sums, suppose that X and Y are independent random variables with log-concave distribution. Since the product of two log-concave functions is log-concave, the joint distribution of (X,Y) is also log-concave. Log-concavity is preserved by affine changes of coordinates, so the distribution of (X + Y, X â Y) is log-concave as well. Since the distribution of X+Y is a marginal over the joint distribution of (X + Y, X â Y), we conclude that X + Y has a log-concave distribution.
The PrĂ©kopaâLeindler inequality can be used to prove results about concentration of measure.
Theorem[ citation needed] Let , and set . Let denote the standard Gaussian pdf, and its associated measure. Then .
Proof of concentration of measure
|
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The proof of this theorem goes by way of the following lemma: Lemma In the notation of the theorem, . This lemma can be proven from PrĂ©kopaâLeindler by taking and . To verify the hypothesis of the inequality, , note that we only need to consider , in which case . This allows us to calculate: Since , the PL-inequality immediately gives the lemma. To conclude the concentration inequality from the lemma, note that on , , so we have . Applying the lemma and rearranging proves the result. |
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cite book}}
: CS1 maint: location missing publisher (
link)