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


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

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


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

This notation allows the following essential form of the Prékopa–Leindler inequality: let 0 &lt; 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 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 &lt; λ &lt; 1 and A and B are bounded, measurable subsets of Rn such that the Minkowski sum (1 &minus; 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 &lt; λ < 1 and A and B are non-empty, bounded, measurable subsets of Rn such that (1 &minus; λ)A + λB is also measurable, then


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

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) ∈ Rm × Rn, so that by definition we have

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 ($$) satisfies condition ($$) with h(x) = H(x,(1 &minus; 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.

Applications to concentration of measure
The Prékopa–Leindler inequality can be used to prove results about concentration of measure.

Theorem Let $ A \subseteq \mathbb{R}^n $, and set $ A_{\epsilon} = \{ x : d(x,A) < \epsilon \} $. Let $ \gamma(x) $ denote the standard Gaussian pdf, and $ \mu $  its associated measure. Then $ \mu(A_{\epsilon}) \geq 1 - \frac{ e^{ - \epsilon^2/4}}{\mu(A)} $.

The proof of this theorem goes by way of the following lemma:

Lemma In the notation of the theorem, $ \int_{\mathbb{R}^n} \exp ( d(x,A)^2/4) d\mu \leq 1/\mu(A) $.

This lemma can be proven from Prékopa–Leindler by taking $ h(x) = \gamma(x), f(x) = e^{ \frac{ d(x,A)^2}{4}} \gamma(x), g(x) = 1_A(x) \gamma(x) $  and  $ \lambda = 1/2 $. To verify the hypothesis of the inequality, $ h( \frac{ x + y}{2} ) \geq \sqrt{ f(x) g(y)} $, note that we only need to consider $ y \in A  $ , in which case  $ d(x,A) \leq ||x - y|| $. This allows us to calculate:


 * $$ (2 \pi)^n f(x) g(x) = \exp( \frac{ d(x,A) }{4} - ||x||^2/2 - ||y||^2/2 ) \leq \exp( \frac{ ||x - y||^2 }{4} - ||x||^2/2 - ||y||^2/2 ) = \exp ( - ||\frac{x + y}{2}||^2 ) = (2 \pi)^n h( \frac{ x + y}{2})^2. $$

Since $ \int h(x) dx = 1 $, the PL-inequality immediately gives the lemma.

To conclude the concentration inequality from the lemma, note that on $ \mathbb{R}^n \setminus A_{\epsilon} $, $ d(x,A) > \epsilon $ , so we have $ \int_{\mathbb{R}^n} \exp ( d(x,A)^2/4) d\mu \geq ( 1 - \mu(A_{\epsilon})) \exp ( \epsilon^2/4) $. Applying the lemma and rearranging proves the result.