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 Prékopa András and Leindler László.

tatement of the inequality

Let 0 < "λ" < 1 and let "f", "g", "h" : R^"n" → [0, +∞) be non-negative real-valued measurable functions defined on "n"-dimensional Euclidean space R^"n". Suppose that these functions satisfy

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

for all "x" and "y" in R^"n". Then

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

Essential form of the inequality

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

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

This notation allows the following "essential form" of the Prékopa-Leindler inequality: let 0 < "λ" < 1 and let "f", "g" ∈ "L"¹(R^"n"; [0, +∞)) be non-negative absolutely integrable functions. Let

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

Then "s" is measurable and

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

The essential supremum form was given in [cite journal | authors = Herm Jan Brascamp and Elliott H. Lieb | title = 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 = Journal of Functional Analysis | volume = 22 | pages = 366–389 | year = 1976 ] . 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 R^"n" such that the Minkowski sum (1 − "λ")"A" + λ"B" is also measurable, then

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

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 R^"n" such that (1 − "λ")"A" + λ"B" is also measurable, then

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

References

* cite journal
last=Gardner
first=Richard J.
title=The Brunn-Minkowski inequality
journal=Bull. Amer. Math. Soc. (N.S.)
volume=39
issue=3
year=2002
pages = pp. 355–405 (electronic)
issn = 0273-0979
url = http://www.ams.org/bull/2002-39-03/S0273-0979-02-00941-2/S0273-0979-02-00941-2.pdf
doi=10.1090/S0273-0979-02-00941-2