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

$h \left( (1-\lambda)x + \lambda y \right) \geq f(x)^{1 - \lambda} g(y)^\lambda$

(1)

for all x and y in Rⁿ. Then

$\| h\|_{1} := \int_{\mathbb{R}^n} h(x) \, \mathrm{d} x \geq \left( \int_{\mathbb{R}^n} f(x) \, \mathrm{d} x \right)^{1 -\lambda} \left( \int_{\mathbb{R}^n} g(x) \, \mathrm{d} x \right)^\lambda =: \| f\|_1^{1 -\lambda} \| g\|_1^\lambda. \,$

Essential form of the inequality

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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \mathop{\mathrm{ess\,sup}}_{x \in \mathbb{R}^{n}} f(x) = \inf \left\{ t \in [- \infty, + \infty] \mid f(x) \leq t \text{ for almost all } x \in \mathbb{R}^{n} \right\}.

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

$s(x) = \mathop{\mathrm{ess\,sup}}_{y \in \mathbb{R}^n} f \left( \frac{x - y}{1 - \lambda} \right)^{1 - \lambda} g \left( \frac{y}{\lambda} \right)^\lambda.$

Then s is measurable and

$\| s \|_1 \geq \| f \|_1^{1 - \lambda} \| g \|_1^\lambda.$

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

$\mu \left( (1 - \lambda) A + \lambda B \right) \geq \mu (A)^{1 - \lambda} \mu (B)^{\lambda},$

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

$\mu \left( (1 - \lambda) A + \lambda B \right)^{1 / n} \geq (1 - \lambda) \mu (A)^{1 / n} + \lambda \mu (B)^{1 / n}.$

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) ∈ R^m × Rⁿ, so that by definition we have

$H \left( (1 - \lambda)(x_1,y_1) + \lambda (x_2,y_2) \right) \geq H(x_1,y_1)^{1 - \lambda} H(x_2,y_2)^{\lambda},$

(2)

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

$M(y) = \int_{\mathbb{R}^m} H(x,y) \, dx.$

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

$M((1-\lambda) y_1 + \lambda y_2) \geq M(y_1)^{1-\lambda} M(y_2)^\lambda,$

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.