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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Prékopa–Leindler inequality is an
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...
closely related to the reverse Young's inequality, the Brunn–Minkowski inequality and a number of other important and classical inequalities in
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
. The result is named after the Hungarian
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
s
András Prékopa András Prékopa (September 11, 1929 – September 18, 2016) was a Hungarian mathematician, a member of the Hungarian Academy of Sciences. He was one of the pioneers of stochastic programming and has been a major contributor to its literature. H ...
and László Leindler.


Statement of the inequality

Let 0 < ''λ'' < 1 and let ''f'', ''g'', ''h'' : R''n'' →  negative real number">real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an i ...
measurable functions defined on ''n''-dimensional Euclidean space R''n''. Suppose that these functions satisfy for all ''x'' and ''y'' in R''n''. Then :\, h\, _ := \int_ h(x) \, \mathrm x \geq \left( \int_ f(x) \, \mathrm x \right)^ \left( \int_ g(x) \, \mathrm x \right)^\lambda =: \, f\, _1^ \, g\, _1^\lambda.


Essential form of the inequality

Recall that the
essential supremum In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''all' ...
of a measurable function ''f'' : R''n'' → R is defined by :\mathop_ f(x) = \inf \left\. This notation allows the following ''essential form'' of the Prékopa–Leindler inequality: let 0 < ''λ'' < 1 and let ''f'', ''g'' ∈ ''L''1(R''n''; 
absolutely integrable In mathematics, an absolutely integrable function is a function whose absolute value is integrable, meaning that the integral of the absolute value over the whole domain is finite. For a real-valued function, since \int , f(x), \, dx = \int f^+ ...
functions. Let :s(x) = \mathop_ f \left( \frac \right)^ g \left( \frac \right)^\lambda. Then ''s'' is measurable and :\"> s \, _1 \geq \, f \, _1^ \, g \, _1^\lambda. The essential supremum form was given by Herm Brascamp and

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 Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded e ...
, measurable subsets of R''n'' such that the Minkowski sum">measurable set">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 \right) \geq \mu (A)^ \mu (B)^, 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-
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,
bounded Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded e ...
, measurable subsets of R''n'' such that (1 − ''λ'')''A'' + λ''B'' is also measurable, then :\mu \left( (1 - \lambda) A + \lambda B \right)^ \geq (1 - \lambda) \mu (A)^ + \lambda \mu (B)^.


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 Social exclusion or social marginalisation is the social disadvantage and relegation to the fringe of society. It is a term that has been used widely in Europe and was first used in France in the late 20th century. It is used across discipline ...
and
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...
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 and let ''M''(''y'') denote the marginal distribution obtained by integrating over ''x'': :M(y) = \int_ H(x,y) \, dx. Let ''y''1, ''y''2 ∈ R''n'' and 0 < ''λ'' < 1 be given. Then equation () satisfies condition () with ''h''(''x'') = ''H''(''x'',(1 − ''λ'')y1 + ''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)^ 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^n , and set A_ = \ . Let \gamma(x) denote the standard Gaussian pdf, and \mu its associated measure. Then \mu(A_) \geq 1 - \frac . The proof of this theorem goes by way of the following lemma: Lemma In the notation of the theorem, \int_ \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^ \gamma(x), g(x) = 1_A(x) \gamma(x) and \lambda = 1/2 . To verify the hypothesis of the inequality, h( \frac ) \geq \sqrt , 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 - , , x, , ^2/2 - , , y, , ^2/2 ) \leq \exp( \frac - , , x, , ^2/2 - , , y, , ^2/2 ) = \exp ( - , , \frac, , ^2 ) = (2 \pi)^n h( \frac)^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^n \setminus A_ , d(x,A) > \epsilon , so we have \int_ \exp ( d(x,A)^2/4) d\mu \geq ( 1 - \mu(A_)) \exp ( \epsilon^2/4) . Applying the lemma and rearranging proves the result.


References


Further reading

* * {{DEFAULTSORT:Prekopa-Leindler Inequality Geometric inequalities Integral geometry Real analysis Theorems in analysis Theorems in measure theory