## Isoperimetry between exponential and Gaussian

Venue: | Electronic J. Prob |

Citations: | 15 - 7 self |

### BibTeX

@ARTICLE{Barthe_isoperimetrybetween,

author = {F. Barthe and P. Cattiaux and C. Roberto},

title = {Isoperimetry between exponential and Gaussian},

journal = {Electronic J. Prob},

year = {},

pages = {2346509}

}

### OpenURL

### Abstract

We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the isoperimetric problem. 1

### Citations

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288 | Concentration of measure and isoperimetric inequalities in product spaces
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- 1995
(Show Context)
Citation Context ... −1 (γ n (A)). These inequalities are best possible, hence Iγ n = G′ ◦G −1 is independent of the dimension n. Such dimension free properties are crucial in the study of large random systems, see e.g. =-=[27, 37]-=-. Asking which measures enjoy such a dimension free isoperimetric inequality is therefore a fundamental question. Let us be more specific about the products we are considering: if µ is a probability m... |

174 |
Sobolev Spaces
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(Show Context)
Citation Context ...uch that for all A ⊂ Ω with µ(A) < 1/2 one has B Cap µ(A) ≥ Φ(1IA). Then for every smooth function f : M → R vanishing on Ωc it holds Φ(f 2 ∫ ) ≤ 4B |∇f| 2 dµ. Proof. We start with a result of Maz’ja =-=[29]-=-, also discussed in [7, Proposition 13]: given two absolutely continuous measures µ, ν on M, denote by Bν the smallest constant such that for all A ⊂ Ω one has Ω BνCap µ (A, Ω) ≥ ν(A). X } . 6Then fo... |

126 |
Sur les inégalités de Sobolev logarithmiques, Panoramas et Synthèses [Panoramas
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- 2000
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Citation Context ... from Corollary 8 (the hypotheses on Φ ensure that T is non-decreasing and T(x)/x is non-increasing). Example 1. A first family of examples is given by the measures dµp(x) = e−|x|pdx/(2Γ(1+1/p)), p ∈ =-=[1, 2]-=-. The potential x ↦→ |x| p fulfills the hypotheses of Proposition 9 with Tp(x) = 1 1 2(1− p p2x ) . Thus, by Proposition 9, for any n ≥ 1, 9µ n p satisfies a super Poincaré inequality with function β... |

103 |
Concentration of measure and logarithmic Sobolev inequalities. Séminaire de
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- 1999
(Show Context)
Citation Context ... −1 (γ n (A)). These inequalities are best possible, hence Iγ n = G′ ◦G −1 is independent of the dimension n. Such dimension free properties are crucial in the study of large random systems, see e.g. =-=[27, 37]-=-. Asking which measures enjoy such a dimension free isoperimetric inequality is therefore a fundamental question. Let us be more specific about the products we are considering: if µ is a probability m... |

100 |
The Brunn-Minkowski inequality in Gauss Spaces
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- 1974
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Citation Context ...clidean space (R n , | · |) with the standard Gaussian measure, denoted γ n in order to emphasize its product structure dx dγ n (x) = e −|x|2 /2 (2π) n/2 , x ∈ Rn . Sudakov-Tsirel’son [35] and Borell =-=[18]-=- have shown that among sets of prescribed measure, half-spaces have h-enlargements of minimal measure. Setting G(t) = γ((−∞, t]), their result reads as follows: for A ⊂ Rn set a = G−1 (γn(A)), then γ ... |

98 | Exponential integrability and transportation cost related to logarithmic Sobolev inequalities
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- 1999
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Citation Context ...t constant such that for all A ⊂ Ω one has Ω BνCap µ (A, Ω) ≥ ν(A). X } . 6Then for every smooth function f : M → R vanishing on Ω c ∫ f 2 dν ≤ 4Bν ∫ |∇f| 2 dµ. Following an idea of Bobkov and Götze =-=[12]-=- we apply the previous inequality to the measures dν = gdµ for g ∈ G. Thus for f as above ∫ ∫ Φ(f) = sup |∇f| g∈G 2 dµ. Ω fg dµ ≤ sup Bg dµ g∈G It remains to check that the constant B is at most sup g... |

91 | and Michel Émery. Diffusions hypercontractives - Bakry |

75 |
The concentration of measure phenomenon, volume 89 ofMathematical Surveys and Monographs
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Citation Context ... 1 − Rh(1/2) is the so-called concentration function, the isoperimetric problem for probability measures is closely related to the concentration of measure phenomenon. We refer the reader to the book =-=[28]-=- for more details on this topic. 1The main probabilistic example where the isoperimetric problem is completely solved is the Euclidean space (R n , | · |) with the standard Gaussian measure, denoted ... |

49 |
Logarithmic Sobolev inequalities and stochastic Ising models
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- 1987
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Citation Context ...he picture by studying how perturbations affect our inequalities. In this section µ is non-negative measure, and ν = e −2V µ is a probability measure. Recall a well known result by Holley and Stroock =-=[23]-=-: if a probability measure µ satisfies a logarithmic Sobolev inequality with constant C then ν satisfies a logarithmic Sobolev inequality with constant at most Ce Osc(2V ) , where Osc(V ) = supV − min... |

41 |
Between Sobolev and Poincaré, Geometric Aspects of Functional Analysis
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Citation Context ...é inequalities imply precise isoperimetric estimates, and are related to Beckner-type inequalities via F-Sobolev inequalities. In fact, Beckner-type inequalities, as developed by Lata̷la-Oleszkiewicz =-=[25]-=- were crucial in deriving dimension-free concentration in our paper [7]. In full generality they read as follows: for all smooth f and all p ∈ [1, 2), ∫ f 2 (∫ dµ − |f| p ) 2 ∫ p dµ ≤ T(2 − p) |∇f| 2 ... |

38 |
Lévy-Gromov’s isoperimetric inequality for an infinitedimensional diffusion generator
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Citation Context ...n phenomenon and isoperimetric profile for probability measures, in the intermediate regime between exponential and Gaussian. Our approach of the isoperimetric inequality followed the 2one of Ledoux =-=[3]-=-: we studied the improving properties of the underlying semigroups, but we had to replace Gross hypercontractivity by a notion of Orlicz hyperboundedness, closely related to F-Sobolev inequalities. Th... |

37 | Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
- Barthe, Cattiaux, et al.
(Show Context)
Citation Context ...he exponential measure, where the enlargements involve mixtures of ℓ1 and ℓ2 balls with different scales (this result does not provide lower bounds on the boundary measure of sets). In a recent paper =-=[8]-=- we have studied in depth various types of inequalities allowing the precise description of concentration phenomenon and isoperimetric profile for probability measures, in the intermediate regime betw... |

30 | Functional inequalities for empty essential spectrum
- Wang
(Show Context)
Citation Context ...general measures, at the price of rather heavy technicalities. In this paper, we wish to point out a softer approach to isoperimetric inequalities. It was recently developed by Wang and his coauthors =-=[38, 21, 39]-=- and relies on so called super-Poincaré inequalities. It can be combined with our techniques in order to provide dimension free isoperimetric inequalities for large classes of measures. Among them are... |

29 |
Uniformly positivity improving property, Sobolev inequalities and spectral gaps
- Aida
(Show Context)
Citation Context ...mal boundary measure is very difficult. In many cases the only hope is to estimate the isoperimetric function of the metric measured space (M, d, µ), denoted by Iµ Iµ(a) := inf{µs(∂A); µ(A) = a}, a ∈ =-=[0, 1]-=-. For h > 0 one may also investigate the best function Rh such that µ(Ah) ≥ Rh(µ(A)) holds for all Borel sets. The two questions are related, and even equivalent in simple situations, see [16]. Since ... |

29 |
A new isoperimetric inequality and the concentration of measure phenomenon, Geom. Aspects of Funct. Anal
- Talagrand
- 1991
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Citation Context ...ed from below by a continuous positive function on (0, 1), standard arguments imply that the measures µ n all satisfy a concentration inequality which is independent of n. As observed by Talagrand in =-=[40]-=-, this property implies the existence of ε > 0 such that ∫ e ε|t| dµ(t) < +∞, see [16] for more precise results. In particular, the central limit theorem applies to µ. Setting m = ∫ xdµ(x), it allows ... |

28 |
Sobolev inequalities for probability measures on the real line
- Barthe, Roberto
(Show Context)
Citation Context ... [7]. In full generality they read as follows: for all smooth f and all p ∈ [1, 2), ∫ f 2 (∫ dµ − |f| p ) 2 ∫ p dµ ≤ T(2 − p) |∇f| 2 dµ, where T : (0, 1] → R + is a non-decreasing function. Following =-=[8]-=- we could characterize the measures on the line which enjoy this property, and then take advantage of the tensorization property. As the reader noticed, the super-Poincaré and Beckner-type inequalitie... |

27 | Isoperimetric constants for product probability measures
- Bobkov, Houdré
- 1997
(Show Context)
Citation Context ... show that measures on the real line with this property must have at least Gaussian tails and at most exponential tails. For the symmetric exponential law dν(t) = e−|t| dt/2, t ∈ R, Bobkov and Houdré =-=[14]-=- actually showed Iν∞ ≥ Iν/(2 √ 6). Their argument uses a functional isoperimetric inequality with the tensorization property. In an earlier paper, Talagrand [36] proved a different dimension free isop... |

26 |
Entropy bounds and isoperimetry
- Bobkov, Zegarlinski
(Show Context)
Citation Context ...Φ−1 (log 1 = 1. t ) Consequently, if Φ(0) < log 2, LΦ is defined on [0, 1] and there exists constants k1, k2 > 0 such that for all t ∈ [0, 1], k1LΦ(t) ≤ IΦ(t) ≤ k2LΦ(t). ) . 12This result appears in =-=[4, 17]-=- in the particular case Φ(x) = |x| p . Proof. Since Φ is convex and (strictly) increasing, note that Φ ′ may vanish only at 0. Under our assumptions on Φ we have H(y) = ∫ y −∞ Z−1 Φ e−Φ(|x|) dx ∼ Z −1... |

22 |
Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups
- Röckner, Wang
(Show Context)
Citation Context ...s section we collect results which relate super-Poincaré inequalities with isoperimetry. They follow Ledoux approach of Buser’s inequality [26]. This method was developed by Bakry-Ledoux [3] and Wang =-=[32, 38]-=-, see also [7]. The following result, a particular case of [3, Inequality (4.3)], allows to derive isoperimetric estimates from semi-group bounds. Theorem 10 ([3]). Let µ be a probability measure on (... |

21 | Hypercontractivity for perturbed diffusion semi-groups
- Cattiaux
- 2005
(Show Context)
Citation Context ...n the spectral analysis of the Fokker-Planck operator and of its analogue for particle systems. This correspondence can be explained via Ito’s stochastic calculus and the Girsanov transform, see e.g. =-=[19]-=-. The following result is a consequence of Theorem 3 and Example 14 in [7]. We recall that a Young function Φ satifies the ∆2 condition if Φ(2x) ≤ KΦ(x) for some K > 1 and x ≥ x0 ≥ 0; and the ∇2 condi... |

18 |
Extremal properties of half-spaces for log-concave distributions
- Bobkov
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(Show Context)
Citation Context ...Φ(dx) = Z −1 Φ exp{−Φ(|x|)}dx = ϕ(x)dx, x ∈ R, with Φ convex and √ Φ concave. The isoperimetric profile of a symmetric log-concave density on the line (with the usual metric) was calculated by Bobkov =-=[10]-=-. He showed that half-lines have minimal boundary among sets of given measure. Since the boundary measure of (−∞, x] is given by the density of the measure at x, the isoperimetric profile is IΦ(t) = ϕ... |

18 |
Some connections between isoperimetric and Sobolev-type inequalities
- Bobkov, Houdré
- 1997
(Show Context)
Citation Context ..., a ∈ [0, 1]. For h > 0 one may also investigate the best function Rh such that µ(Ah) ≥ Rh(µ(A)) holds for all Borel sets. The two questions are related, and even equivalent in simple situations, see =-=[16]-=-. Since the function α(h) = 1 − Rh(1/2) is the so-called concentration function, the isoperimetric problem for probability measures is closely related to the concentration of measure phenomenon. We re... |

18 |
The isoperimetric problem. In: Global theory of minimal surfaces
- Ros
(Show Context)
Citation Context ...by µs(∂A) := liminf h→0 + µ(Ah \ A) · h An isoperimetric inequality is a lower bound on the boundary measure of sets in terms of their measure. Their study is an important topic in geometry, see e.g. =-=[33]-=-. Finding sets of minimal boundary measure is very difficult. In many cases the only hope is to estimate the isoperimetric function of the metric measured space (M, d, µ), denoted by Iµ Iµ(a) := inf{µ... |

17 |
A simple analytic proof of an inequality by
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(Show Context)
Citation Context ... 1. (log log(e + s)) 2α/p 4 Isoperimetric inequalities In this section we collect results which relate super-Poincaré inequalities with isoperimetry. They follow Ledoux approach of Buser’s inequality =-=[26]-=-. This method was developed by Bakry-Ledoux [3] and Wang [32, 38], see also [7]. The following result, a particular case of [3, Inequality (4.3)], allows to derive isoperimetric estimates from semi-gr... |

16 | Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities - Rothaus - 1985 |

14 | Coercive inequalities for Gibbs measures
- Bertini, Zegarlinski
(Show Context)
Citation Context ...ng on s turns this family of inequalities into a single one, which belongs to Nash inequalities. But it is often easier to work with the first form. Similar inequalities appear in the literature, see =-=[9, 20]-=-. Wang discovered that superPoincaré inequalities imply precise isoperimetric estimates, and are related to Beckner-type inequalities via F-Sobolev inequalities. In fact, Beckner-type inequalities, as... |

14 |
Heat kernels and spectral theory, volume 92 of Cambridge Tracts in Mathematics
- Davies
- 1990
(Show Context)
Citation Context ...ng on s turns this family of inequalities into a single one, which belongs to Nash inequalities. But it is often easier to work with the first form. Similar inequalities appear in the literature, see =-=[9, 20]-=-. Wang discovered that superPoincaré inequalities imply precise isoperimetric estimates, and are related to Beckner-type inequalities via F-Sobolev inequalities. In fact, Beckner-type inequalities, as... |

13 | Functional inequalities for uniformly integrable semigroups and applications
- Gong, Wang
(Show Context)
Citation Context ...general measures, at the price of rather heavy technicalities. In this paper, we wish to point out a softer approach to isoperimetric inequalities. It was recently developed by Wang and his coauthors =-=[38, 21, 39]-=- and relies on so called super-Poincaré inequalities. It can be combined with our techniques in order to provide dimension free isoperimetric inequalities for large classes of measures. Among them are... |

12 | Orlicz-Sobolev inequalities for sub-gaussian measures and ergodicity of Markov semi-groups
- Roberto, Zegarlinski
- 2005
(Show Context)
Citation Context ...n ) ≤ DφCap µ n(A), (A) for some constant Dφ (independent on n) and all A with µ n (A) ≤ 1/2. This achieves the proof. Remark 6. Part of the previous Theorem is proved in a slightly different form in =-=[31]-=-. Remark 7. The condition F(8) > 0 can be relaxed just changing F(x) in F(ρx) for a large enough ρ and arguing as before it will only modify the constant Cφ. Remark 8. In the capacity-measure inequali... |

10 |
Log-concave and spherical models in isoperimetry, Geom
- Barthe
(Show Context)
Citation Context ... comparison result will allow us to modify measures without loosing much on their isoperimetric profile. It also shows that even log-concave measures on the real line play a central role. Theorem 14 (=-=[5, 33]-=-). Let m be a probability measure on (R, |.|) with even logconcave density. Let µ be a probability measure on (M, g) such that Iµ ≥ cIm. Then for all n ≥ 1, Iµ n ≥ cImn. c √ cy 13Now we show the foll... |

9 |
Levels of concentration between exponential and
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- 2001
(Show Context)
Citation Context ...Φ−1 (log 1 = 1. t ) Consequently, if Φ(0) < log 2, LΦ is defined on [0, 1] and there exists constants k1, k2 > 0 such that for all t ∈ [0, 1], k1LΦ(t) ≤ IΦ(t) ≤ k2LΦ(t). ) . 12This result appears in =-=[4, 17]-=- in the particular case Φ(x) = |x| p . Proof. Since Φ is convex and (strictly) increasing, note that Φ ′ may vanish only at 0. Under our assumptions on Φ we have H(y) = ∫ y −∞ Z−1 Φ e−Φ(|x|) dx ∼ Z −1... |

6 |
Weak dimension-free concentration of measure
- Bobkov, Houdré
(Show Context)
Citation Context ... on the product Riemannian manifold M n where the geodesic distance is the ℓ2 combination of the distances on the factors. Considering the ℓ∞ combination is easier and leads to different results, see =-=[11, 15, 6]-=-. It can be shown that Gaussian measures on the line are the only ones for which coordinate half-spaces {x1 ≤ t} solve the isoperimetric problem in any dimension [13, 24, 30]. Therefore it is natural ... |

5 |
Infinite dimensional isoperimetric inequalities in product spaces with the supremum distance
- Barthe
(Show Context)
Citation Context ... on the product Riemannian manifold M n where the geodesic distance is the ℓ2 combination of the distances on the factors. Considering the ℓ∞ combination is easier and leads to different results, see =-=[11, 15, 6]-=-. It can be shown that Gaussian measures on the line are the only ones for which coordinate half-spaces {x1 ≤ t} solve the isoperimetric problem in any dimension [13, 24, 30]. Therefore it is natural ... |

5 |
Tsirel’son, Extremal propreties of halfspaces for spherically invariant measures
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- 1978
(Show Context)
Citation Context ...solved is the Euclidean space (R n , | · |) with the standard Gaussian measure, denoted γ n in order to emphasize its product structure dx dγ n (x) = e −|x|2 /2 (2π) n/2 , x ∈ Rn . Sudakov-Tsirel’son =-=[35]-=- and Borell [18] have shown that among sets of prescribed measure, half-spaces have h-enlargements of minimal measure. Setting G(t) = γ((−∞, t]), their result reads as follows: for A ⊂ Rn set a = G−1 ... |

4 |
Characterization of Gaussian measures in terms of the isoperimetric properties of half-spaces
- Bobkov, Houdré
- 1996
(Show Context)
Citation Context ... to different results, see [11, 15, 6]. It can be shown that Gaussian measures on the line are the only ones for which coordinate half-spaces {x1 ≤ t} solve the isoperimetric problem in any dimension =-=[13, 24, 30]-=-. Therefore it is natural to investigate measures for which one dimensional sets are approximate solutions of the isoperimetric problem for the products, that is for which there exists c > 0 with Iµ ≥... |

4 |
Intrinsic bounds on some real-valued stationary random functions
- Borell
- 1985
(Show Context)
Citation Context ...Φ(dx) = Z −1 Φ exp{−Φ(|x|)}dx = ϕ(x)dx, x ∈ R, with Φ convex and √ Φ concave. The isoperimetric profile of a symmetric log-concave density on the line (with the usual metric) was calculated by Borell =-=[20]-=- (see also Bobkov [11]). He showed that half-lines have minimal boundary among sets of given measure. Since the boundary measure of (−∞, x] is given by the density of the measure at x, the isoperimetr... |

2 |
Isoperimetric problem for uniform enlargement
- Bobkov
- 1997
(Show Context)
Citation Context ... on the product Riemannian manifold M n where the geodesic distance is the ℓ2 combination of the distances on the factors. Considering the ℓ∞ combination is easier and leads to different results, see =-=[11, 15, 6]-=-. It can be shown that Gaussian measures on the line are the only ones for which coordinate half-spaces {x1 ≤ t} solve the isoperimetric problem in any dimension [13, 24, 30]. Therefore it is natural ... |

2 |
A proof of a conjecture of Bobkov and Houdré
- Kwapien, Pycia, et al.
- 1996
(Show Context)
Citation Context ... to different results, see [11, 15, 6]. It can be shown that Gaussian measures on the line are the only ones for which coordinate half-spaces {x1 ≤ t} solve the isoperimetric problem in any dimension =-=[13, 24, 30]-=-. Therefore it is natural to investigate measures for which one dimensional sets are approximate solutions of the isoperimetric problem for the products, that is for which there exists c > 0 with Iµ ≥... |

2 |
A generalization of poincaré and log-sobolev inequalities. Potential Analysis, 22:1–15, 2005. Université Paris-Est - Laboratoire d’Analyse et de
- Wang
(Show Context)
Citation Context ...general measures, at the price of rather heavy technicalities. In this paper, we wish to point out a softer approach to isoperimetric inequalities. It was recently developed by Wang and his coauthors =-=[38, 21, 39]-=- and relies on so called super-Poincaré inequalities. It can be combined with our techniques in order to provide dimension free isoperimetric inequalities for large classes of measures. Among them are... |

1 |
On certain characterization of normal distribution
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- 1997
(Show Context)
Citation Context ... to different results, see [11, 15, 6]. It can be shown that Gaussian measures on the line are the only ones for which coordinate half-spaces {x1 ≤ t} solve the isoperimetric problem in any dimension =-=[13, 24, 30]-=-. Therefore it is natural to investigate measures for which one dimensional sets are approximate solutions of the isoperimetric problem for the products, that is for which there exists c > 0 with Iµ ≥... |

1 |
Hypercontractivité et isopérimétrie gaussienne. Applications aux systèmes de spins
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- 2000
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Citation Context ...t results which relate super-Poincaré inequalities with isoperimetry. They follow Ledoux approach of Buser’s inequality [30]. This method was developed by Bakry-Ledoux [4] and Wang [36, 42], see also =-=[24, 8]-=-. 10The following result, a particular case of [4, Inequality (4.3)], allows to derive isoperimetric estimates from semi-group bounds. Theorem 10 ([4, 36]). Let µ be a probability measure on (M, g) w... |

1 | statistical mechanics. Series on Partial Differential Equations and Applications - analysis, Laplacians - 1987 |