Prékopa–Leindler inequality
In mathematics, the Prékopa–Leindler inequality is an integral inequality closely related to the reverse Young's inequality, the Brunn–Minkowski inequality and a number of other important and classical inequalities in analysis. The result is named after the Hungarian mathematicians András Prékopa and László Leindler.
Statement of the inequality
Let 0 < λ < 1 and let f, g, h : Rn → [0, +∞) 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
Essential form of the inequality
Recall that the essential supremum of a measurable function f : Rn → R is defined by
This notation allows the following essential form of the Prékopa–Leindler inequality: let 0 < λ < 1 and let f, g ∈ L1(Rn; [0, +∞)) be non-negative absolutely integrable functions. Let
Then s is measurable and
The essential supremum form was given in.[1] 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.
Relationship to the Brunn–Minkowski inequality
It can be shown that the usual Prékopa–Leindler inequality implies the Brunn–Minkowski 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 μ denotes n-dimensional Lebesgue measure. Hence, the Prékopa–Leindler inequality can also be used[2] to prove the Brunn–Minkowski 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
Applications in probability and statistics
Log-concave distributions
The Prékopa–Leindler 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. Suppose that H(x,y) is a log-concave distribution for (x,y) ∈ Rm × Rn, so that by definition we have
(2) |
and let M(y) denote the marginal distribution obtained by integrating over x:
Let y1, y2 ∈ 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 Prékopa–Leindler 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 − Y) is log-concave as well. Since the distribution of X+Y is a marginal over the joint distribution of (X + Y, X − Y), we conclude that X + Y has a log-concave distribution.
Applications to concentration of measure
The Prékopa–Leindler 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 Prékopa–Leindler 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. |
Notes
- ^ "On extensions of the Brunn–Minkowski and Prekopa–Leindler theorems, including inequalities for log concave functions and with an application to the diffusion equation". Journal of Functional Analysis. 22 (4): 366–389. 1976. doi:10.1016/0022-1236(76)90004-5.
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ignored (help) - ^ Gardner, Richard J. (2002). "The Brunn–Minkowski inequality". Bull. Amer. Math. Soc. (N.S.) 39 (3): pp. 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2. ISSN 0273-0979.
References
- Gardner, Richard J. (2002). "The Brunn–Minkowski inequality" (PDF). Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2. ISSN 0273-0979.
- Prékopa, András (1971). "Logarithmic concave measures with application to stochastic programming" (PDF). Acta Sci. Math. 32: 301–316.
- Prékopa, András (1973). "On logarithmic concave measures and functions" (PDF). Acta Sci. Math. 34: 335–343.