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 := ∫ R n h ( x ) d x ≥ ( ∫ R n f ( x ) d x ) 1 − λ ( ∫ R n g ( x ) d x ) λ =: ‖ f ‖ 1 1 − λ ‖ g ‖ 1 λ . {\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 }.}
Essential form of the inequality Recall that the essential supremum of a measurable function f : Rn → R is defined by
e s s s u p x ∈ R n f ( x ) = inf { t ∈ [ − ∞ , + ∞ ] ∣ f ( x ) ≤ t for almost all x ∈ R n } . {\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
s ( x ) = e s s s u p y ∈ R n f ( x − y 1 − λ ) 1 − λ g ( y λ ) λ . {\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
‖ s ‖ 1 ≥ ‖ f ‖ 1 1 − λ ‖ g ‖ 1 λ . {\displaystyle \|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 Rn such that the Minkowski sum (1 − λ)A + λB is also measurable, then
μ ( ( 1 − λ ) A + λ B ) ≥ μ ( A ) 1 − λ μ ( B ) λ , {\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 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
μ ( ( 1 − λ ) A + λ B ) 1 / n ≥ ( 1 − λ ) μ ( A ) 1 / n + λ μ ( B ) 1 / n . {\displaystyle \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 {\displaystyle X,Y} have pdf f , g {\displaystyle f,g} , and X , Y {\displaystyle X,Y} are independent, then f ⋆ g {\displaystyle f\star g} is the pdf of X + Y {\displaystyle 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 ) = ∫ R m H ( x , y ) d 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
M ( ( 1 − λ ) y 1 + λ y 2 ) ≥ M ( y 1 ) 1 − λ M ( y 2 ) λ , {\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.
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 − ϵ 2 / 4 μ ( A ) {\textstyle \mu (A_{\epsilon })\geq 1-{\frac {e^{-\epsilon ^{2}/4}}{\mu (A)}}} .
References
Further reading Eaton, Morris L. (1987). "Log concavity and related topics". Lectures on Topics in Probability Inequalities. Amsterdam. pp. 77–109. ISBN 90-6196-316-8.{{cite book}}: CS1 maint: location missing publisher (link) Wainwright, Martin J. (2019). "Concentration of Measure". High-Dimensional Statistics: A Non-Asymptotic Viewpoint. Cambridge University Press. pp. 72–76. ISBN 978-1-108-49802-9.