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
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) be non-negative absolutely integrable functions. Let
Then s is measurable and
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 A + λB is also measurable, then
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 A + λB is also measurable, then

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 is a log-concave distribution for ∈ Rm × Rn, so that by definition we have
and let M denote the marginal distribution obtained by integrating over x:
Let y1, y2 ∈ Rn and 0 < λ < 1 be given. Then equation satisfies condition with h = H, f = H and g = H, 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 is also log-concave. Log-concavity is preserved by affine changes of coordinates, so the distribution of is log-concave as well. Since the distribution of X+Y is a marginal over the joint distribution of, we conclude that X + Y has a log-concave distribution.