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Prékopa–Leindler inequality

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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.

Contents

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

for all x and y in Rn. Then

∥ h ∥ 1 := ∫ R n h ( x ) d x ≥ ( ∫ R n f ( x ) d x ) 1 − λ ( ∫ R n g ( x ) d x ) λ =: ∥ f ∥ 1 1 − λ ∥ g ∥ 1 λ .

Essential form of the inequality

Recall that the essential supremum of a measurable function f : Rn → R is defined by

e s s s u p x ∈ R n f ( x ) = inf { t ∈ [ − ∞ , + ∞ ] ∣ f ( x ) ≤ t  for almost all  x ∈ R n } .

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

s ( x ) = e s s s u p y ∈ R n f ( x − y 1 − λ ) 1 − λ g ( y λ ) λ .

Then s is measurable and

∥ s ∥ 1 ≥ ∥ f ∥ 1 1 − λ ∥ g ∥ 1 λ .

The essential supremum form was given in. 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

μ ( ( 1 − λ ) A + λ B ) ≥ μ ( A ) 1 − λ μ ( B ) λ ,

where μ denotes n-dimensional Lebesgue measure. Hence, the Prékopa–Leindler inequality can also be used 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

μ ( ( 1 − λ ) A + λ B ) 1 / n ≥ ( 1 − λ ) μ ( A ) 1 / n + λ μ ( B ) 1 / n .

Applications in probability and statistics

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

and let M(y) denote the marginal distribution obtained by integrating over x:

M ( y ) = ∫ R m H ( x , y ) d 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

M ( ( 1 − λ ) y 1 + λ y 2 ) ≥ M ( y 1 ) 1 − λ M ( y 2 ) λ ,

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.

References

Prékopa–Leindler inequality Wikipedia
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