Prékopa–Leindler 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

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

(1)

for all x and y in Rⁿ. Then

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

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

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

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

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

Then s is measurable and

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

The essential supremum form was given in.[1] 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.

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

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

where μ denotes n-dimensional Lebesgue measure. Hence, the Prékopa–Leindler inequality can also be used[2] 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

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

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

${\displaystyle H\left((1-\lambda )(x_{1},y_{1})+\lambda (x_{2},y_{2})\right)\geq H(x_{1},y_{1})^{1-\lambda }H(x_{2},y_{2})^{\lambda },}$

(2)

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

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

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

${\displaystyle M((1-\lambda )y_{1}+\lambda y_{2})\geq M(y_{1})^{1-\lambda }M(y_{2})^{\lambda },}$

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.

• Gardner, Richard J. (2002). "The Brunn–Minkowski inequality" (PDF). Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2. ISSN 0273-0979.

• Prékopa, András (1971). "Logarithmic concave measures with application to stochastic programming" (PDF). Acta Sci. Math. 32: 301–316.

• Prékopa, András (1973). "On logarithmic concave measures and functions" (PDF). Acta Sci. Math. 34: 335–343.