{"history":[],"article":{"id":"b4137079-00ff-49a5-8ef9-ea43cf1424f4","title":"Prékopa–Leindler inequality","slug":"pr-kopa-leindler-inequality","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **Prékopa–Leindler [inequality](https://bliptext.com/articles/inequality-mathematics)** is an [integral](https://bliptext.com/articles/integral) [inequality](https://bliptext.com/articles/inequality-mathematics) closely related to the reverse Young's [inequality](https://bliptext.com/articles/inequality-mathematics), the Brunn–Minkowski [inequality](https://bliptext.com/articles/inequality-mathematics) and a number of other important and classical inequalities in [analysis](https://bliptext.com/articles/mathematical-analysis).\nThe result is named after the [Hungarian](https://bliptext.com/articles/hungary) [mathematicians](https://bliptext.com/articles/mathematician) [András Prékopa](https://bliptext.com/articles/andr-s-pr-kopa) and [László Leindler](https://bliptext.com/articles/l-szl-leindler).\n\n# Statement of the inequality\nLet 0 < λ < 1 and let f, g, h : ****R****n → [0, +∞) be non-[negative](https://bliptext.com/articles/negative-number) [real-valued](https://bliptext.com/articles/real-number) [measurable functions](https://bliptext.com/articles/measurable-function) defined on n-dimensional [Euclidean space](https://bliptext.com/articles/euclidean-space) ****R****n. Suppose that these functions satisfy\nfor all x and y in **R**n. Then\n\n# Essential form of the inequality\n****R****ecall that the [essential supremum](https://bliptext.com/articles/essential-supremum) of a measurable function f : ****R****n → ****R**** is defined by\nThis notation allows the following essential form of the Prékopa–Leindler inequality: let 0 < λ < 1 and let f, g ∈ L1(**R**n; [0, +∞)) be non-negative [absolutely integrable](https://bliptext.com/articles/absolutely-integrable-function) functions. Let\nThen s is measurable and\nThe essential supremum form was given by Herm Brascamp and [Elliott Lieb](https://bliptext.com/articles/elliott-lieb).\nIts use can change the left side of the inequality.\nFor 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.\n\n# Relationship to the Brunn–Minkowski inequality\nIt can be shown that the usual Prékopa–Leindler inequality implies the [Brunn–Minkowski inequality](https://bliptext.com/articles/brunn-minkowski-inequality) in the following form: if 0 < λ < 1 and A and B are [bounded](https://bliptext.com/articles/bounded-set), [measurable subsets](https://bliptext.com/articles/measurable-set) of **R**n such that the [Minkowski sum](https://bliptext.com/articles/minkowski-sum) (1 − λ)A + λB is also measurable, then\nwhere μ denotes n-dimensional [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure).\nHence, 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](https://bliptext.com/articles/empty-set), [bounded](https://bliptext.com/articles/bounded-set), [measurable subsets](https://bliptext.com/articles/measurable-set) of **R**n such that (1 − λ)A + λB is also measurable, then\n\n# Applications in probability and statistics\n\n## Log-concave distributions\nThe Prékopa–Leindler inequality is useful in the theory of [log-concave distributions](https://bliptext.com/articles/logarithmically-concave-function), as it can be used to show that log-concavity is preserved by [marginalization](https://bliptext.com/articles/marginal-distribution) and [independent](https://bliptext.com/articles/independence-probability) summation of log-concave distributed random variables.\nSince, 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.\nSuppose that H(x,y) is a log-concave distribution for (x,y) ∈ ****R****m × ****R****n, so that by definition we have\nand let M(y) denote the marginal distribution obtained by integrating over x:\nLet y1, y2 ∈ **R**n and 0 < λ < 1 be given.\nThen equation satisfies condition 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.\nIt can be written in terms of M as\nwhich is the definition of log-concavity for M.\nTo 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.\nSince the product of two log-concave functions is log-concave, the joint distribution of (X,Y) is also log-concave.\nLog-concavity is preserved by affine changes of coordinates, so the distribution of (X + Y, X − Y) is log-concave as well.\nSince 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.\n\n## Applications to concentration of measure\nThe Prékopa–Leindler inequality can be used to prove results about concentration of measure.\n**Theorem** Let, and set.\nLet \\gamma(x) denote the standard Gaussian pdf, and \\mu its associated measure.\nThen.\nThe proof of this theorem goes by way of the following lemma:\n**Lemma** In the notation of the theorem,.\nThis lemma can be proven from Prékopa–Leindler by taking and.\nTo verify the hypothesis of the inequality,, note that we only need to consider y \\in A, in which case.\nThis allows us to calculate:\nSince, the PL-inequality immediately gives the lemma.\nTo conclude the concentration inequality from the lemma, note that on, , so we have.\nApplying the lemma and rearranging proves the result.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T09:27:49.350Z","updated_at":"2025-01-26T09:27:49.350Z","search_vector":"'inequ':3 'leindler':2 'prékopa':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **Prékopa–Leindler [inequality](https://bliptext.com/articles/inequality-mathematics)** is an [integral](https://bliptext.com/articles/integral) [inequality](https://bliptext.com/articles/inequality-mathematics) closely related to the reverse Young's [inequality](https://bliptext.com/articles/inequality-mathematics), the Brunn–Minkowski [inequality](https://bliptext.com/articles/inequality-mathematics) and a number of other important and classical inequalities in [analysis](https://bliptext.com/articles/mathematical-analysis).\nThe result is named after the [Hungarian](https://bliptext.com/articles/hungary) [mathematicians](https://bliptext.com/articles/mathematician) [András Prékopa](https://bliptext.com/articles/andr-s-pr-kopa) and [László Leindler](https://bliptext.com/articles/l-szl-leindler).\n\n# Statement of the inequality\nLet 0 < λ < 1 and let f, g, h : ****R****n → [0, +∞) be non-[negative](https://bliptext.com/articles/negative-number) [real-valued](https://bliptext.com/articles/real-number) [measurable functions](https://bliptext.com/articles/measurable-function) defined on n-dimensional [Euclidean space](https://bliptext.com/articles/euclidean-space) ****R****n. Suppose that these functions satisfy\nfor all x and y in **R**n. Then\n\n# Essential form of the inequality\n****R****ecall that the [essential supremum](https://bliptext.com/articles/essential-supremum) of a measurable function f : ****R****n → ****R**** is defined by\nThis notation allows the following essential form of the Prékopa–Leindler inequality: let 0 < λ < 1 and let f, g ∈ L1(**R**n; [0, +∞)) be non-negative [absolutely integrable](https://bliptext.com/articles/absolutely-integrable-function) functions. Let\nThen s is measurable and\nThe essential supremum form was given by Herm Brascamp and [Elliott Lieb](https://bliptext.com/articles/elliott-lieb).\nIts use can change the left side of the inequality.\nFor 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.\n\n# Relationship to the Brunn–Minkowski inequality\nIt can be shown that the usual Prékopa–Leindler inequality implies the [Brunn–Minkowski inequality](https://bliptext.com/articles/brunn-minkowski-inequality) in the following form: if 0 < λ < 1 and A and B are [bounded](https://bliptext.com/articles/bounded-set), [measurable subsets](https://bliptext.com/articles/measurable-set) of **R**n such that the [Minkowski sum](https://bliptext.com/articles/minkowski-sum) (1 − λ)A + λB is also measurable, then\nwhere μ denotes n-dimensional [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure).\nHence, 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](https://bliptext.com/articles/empty-set), [bounded](https://bliptext.com/articles/bounded-set), [measurable subsets](https://bliptext.com/articles/measurable-set) of **R**n such that (1 − λ)A + λB is also measurable, then\n\n# Applications in probability and statistics\n\n## Log-concave distributions\nThe Prékopa–Leindler inequality is useful in the theory of [log-concave distributions](https://bliptext.com/articles/logarithmically-concave-function), as it can be used to show that log-concavity is preserved by [marginalization](https://bliptext.com/articles/marginal-distribution) and [independent](https://bliptext.com/articles/independence-probability) summation of log-concave distributed random variables.\nSince, 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.\nSuppose that H(x,y) is a log-concave distribution for (x,y) ∈ ****R****m × ****R****n, so that by definition we have\nand let M(y) denote the marginal distribution obtained by integrating over x:\nLet y1, y2 ∈ **R**n and 0 < λ < 1 be given.\nThen equation satisfies condition 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.\nIt can be written in terms of M as\nwhich is the definition of log-concavity for M.\nTo 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.\nSince the product of two log-concave functions is log-concave, the joint distribution of (X,Y) is also log-concave.\nLog-concavity is preserved by affine changes of coordinates, so the distribution of (X + Y, X − Y) is log-concave as well.\nSince 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.\n\n## Applications to concentration of measure\nThe Prékopa–Leindler inequality can be used to prove results about concentration of measure.\n**Theorem** Let, and set.\nLet \\gamma(x) denote the standard Gaussian pdf, and \\mu its associated measure.\nThen.\nThe proof of this theorem goes by way of the following lemma:\n**Lemma** In the notation of the theorem,.\nThis lemma can be proven from Prékopa–Leindler by taking and.\nTo verify the hypothesis of the inequality,, note that we only need to consider y \\in A, in which case.\nThis allows us to calculate:\nSince, the PL-inequality immediately gives the lemma.\nTo conclude the concentration inequality from the lemma, note that on, , so we have.\nApplying the lemma and rearranging proves the result.","revisionCount":0},"hasMore":false,"nextCursor":null}

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