Prékopa-Leindler inequality

From Wikipedia, the free encyclopedia

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 Prékopa András and Leindler László.

1 Statement of the inequality
2 Essential form of the inequality
3 Relationship to the Brunn-Minkowski inequality
4 References

[edit] 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

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

for all x and y in Rⁿ. Then

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

[edit] Essential form of the inequality

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

$\mathop{\mathrm{ess\,sup}}_{x \in \mathbb{R}^{n}} f(x) = \inf \left\{ t \in [- \infty, + \infty] | f(x) \leq t \mbox{ for almost all } x \in \mathbb{R}^{n} \right\}.$

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

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

[edit] 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 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