# 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 : R$^n$ → [0, +∞) be non-negative real-valued measurable functions defined on n-dimensional Euclidean space R$^n$. Suppose that these functions satisfy
| $h\left((1-\lambda )x+\lambda y\right)\geq f(x)^{1-\lambda }g(y)^{\lambda }$ | (1) |
for all x and y in R$^n$. Then
    $\|h\|_{1}:=\int _{\mathbb {R} ^{n}}h(x)\,\mathrm {d} x\geq \left(\int _{\mathbb {R} ^{n}}f(x)\,\mathrm {d} x\right)^{1-\lambda }\left(\int _{\mathbb {R} ^{n}}g(x)\,\mathrm {d} x\right)^{\lambda }=:\|f\|_{1}^{1-\lambda }\|g\|_{1}^{\lambda }.$

## Essential form of the inequality

Recall that the essential supremum of a measurable function f : R$^n$ → R is defined by
    $\mathop {\mathrm {ess\,sup} } _{x\in \mathbb {R} ^{n}}f(x)=\inf \left\{t\in [-\infty ,+\infty ]\mid f(x)\leq t{\text{ for almost all }}x\in \mathbb {R} ^{n}\right\}.$
This notation allows the following essential form of the Prékopa–Leindler inequality: let 0 < λ < 1 and let f, g ∈ L$^1$(R$^n$; [0, +∞)) be non-negative absolutely integrable functions. Let
    $s(x)=\mathop {\mathrm {ess\,sup} } _{y\in \mathbb {R} ^{n}}f\left({\frac {x-y}{1-\lambda }}\right)^{1-\lambda }g\left({\frac {y}{\lambda }}\right)^{\lambda }.$
Then s is measurable and
    $\|s\|_{1}\geq \|f\|_{1}^{1-\lambda }\|g\|_{1}^{\lambda }.$
The essential supremum form was given by Herm Brascamp and Elliott Lieb. 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 + 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 R$^n$ 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
### Log-concave distributions

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. Since, if $X,Y$ have pdf $f,g$, and $X,Y$ are independent, then $f\star g$ is the pdf of $X+Y$, we also have that the convolution of two log-concave functions is log-concave.
Suppose that H(x,y) is a log-concave distribution for (x,y) ∈ R$^m$ × R$^n$, 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 y$_1$, y$_2$ ∈ R$^n$ and 0 < λ < 1 be given. Then equation (2) satisfies condition (1) with h(x) = H(x,(1 − λ)y$_1$ + λy$_2$), f(x) = H(x,y$_1$) and g(x) = H(x,y$_2$), 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.

### Applications to concentration of measure

The Prékopa–Leindler inequality can be used to prove results about concentration of measure.
Theorem Let A⊆R$^n${\textstyle A\subseteq \mathbb {R} ^{n}}, and set A$_ϵ$={x:d(x, A)<ϵ}{\textstyle A_{\epsilon }=\{x:d(x, A)<\epsilon \}}. Let γ(x){\textstyle \gamma (x)} denote the standard Gaussian pdf, and μ{\textstyle \mu } its associated measure. Then μ(A$_ϵ$)≥1−e$^{−ϵ2/4}$/μ(A) {\textstyle \mu (A_{\epsilon })\geq 1-{\frac {e^{-\epsilon ^{2}/4}}{\mu (A)}}}.