#### DMCA

## 1 Isoperimetry for product of heavy tails distributions

Citations: | 1 - 1 self |

### Citations

336 | A lower bound for the smallest eigenvalue of the Laplacian, in: - Cheeger - 1969 |

201 |
Sur les inegalites de Sobolev logarithmiques, volume 10 of Panoramas et Syntheses. Societe Mathematique de France,
- Ane, Blachere, et al.
- 2000
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Citation Context ... → R+ symmetric around 1/2. Let β(s) = sups≤t≤ 12 t−s J(t) for s ∈ (0, 12 ). Then for all locally Lipschitz functions f : X → R with m(f) = 0,∫ X Φ(f)dµ ≤ ∫ Φ (4β(s)|∇f |) dµ+ 2sOsc(f) ∀s ∈ (0, 1/2). =-=(5)-=- Proof. By the above discussion (see (2)), the assumption µs(∂A) ≥ J (µ(A)) implies that all locally Lipschitz functions f : X → R with m(f) = 0 satisfy∫ X |f |dµ ≤ β(s) ∫ X |∇f |dµ+ sOsc(f) ∀s ∈ (0, ... |

69 |
Isoperimetry and Gaussian analysis,” Lectures on probability theory and statistics,
- Ledoux
- 1996
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Citation Context ...etric space (X, d) equipped with a probability measure µ which is not a Dirac mass at a point. In this note we study the following isoperimetric inequality µs(∂A) ≥ J (µ(A)) A ⊂ X Borel (1) where J : =-=[0, 1]-=- → R+ is symmetric around 1/2 and where the surface measure is defined by the Minkowski content µs(∂A) = lim infε→0 µ(Aε\A) ε with Aε = {x ∈ X : d(x,A) < ε}. For any function f : X → R we define the m... |

50 |
The isoperimetric problem, in “Global theory of minimal surfaces,”
- Ros
- 2005
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Citation Context ...y-ISAAC 3 product spaces, due to the L1 norm of the gradient. Bobkov proposed an alternative functional form of (and equivalent to) the isoperimetric inequalities: I (∫ fdµ ) ≤ ∫ √ I(f)2 + C2|∇f |2dµ =-=(3)-=- where I : [0, 1] → R+ and f : X → [0, 1]. Such inequalities enjoy the tensorisation property and was used by Bobkov26 as an alternative proof of the Gaussian dimension free isoperimetric inequality o... |

45 | On the role of convexity in isoperimetry, spectral gap and concentration, - Milman - 2009 |

36 | Spectral gap, logarithmic Sobolev constant, and geometric bounds. - Ledoux - 2004 |

27 | École Norm - Sci |

17 | On the role of convexity in functional and isoperimetric inequalities - Milman |

16 |
Functional inequalities, Markov processes and Spectral theory
- Wang
- 2005
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Citation Context .... By the above discussion (see (2)), the assumption µs(∂A) ≥ J (µ(A)) implies that all locally Lipschitz functions f : X → R with m(f) = 0 satisfy∫ X |f |dµ ≤ β(s) ∫ X |∇f |dµ+ sOsc(f) ∀s ∈ (0, 1/2). =-=(6)-=- Now consider a bounded function f , locally Lipschitz, with m(f) = 0. Set f+ = max(f, 0) and f− = max(−f, 0). Then m(f+) = m(f−) = 0 and thus m(Φ(f+)) = m(Φ(f−)) = 0. Hence, applying twice (6) to Φ(f... |

16 | Functional inequalities for heavy tailed distributions and application to isoperimetry - Cattiaux, Gozlan, et al. |

5 |
Isoperimetric inequalities, probability measures and convex geometry
- Barthe
- 2005
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Citation Context ...eed, it is easy to prove (see e.g. Ref. 25) that (1) is equivalent to the following weak Cheeger inequality: for any f : X → R locally Lipschitz,∫ |f −m(f)|dµ ≤ β(s) ∫ |∇f |dµ+ s Osc(f) ∀s ∈ (0, 1/2) =-=(2)-=- where m(f) is a the median of f under µ and Osc(f) = sup f − inf f . More precisely (1) implies (2) with β(s) = sups≤t≤ 12 t−s J(t) , and (2) implies (1) with J(t) = sup0<s≤t t−s β(s) for t ∈ (0, 12 ... |

3 | Isoperimetric and concentration inequalities - part i: Equivalence under curvature lower bound. preprint available at arxiv.0902.1560 - Milman - 2009 |

2 | Isoperimetry for product of probability measures: recent results
- Roberto
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Citation Context ...d β(s) = sups≤t≤ 12 t−s J(t) for s ∈ (0, 12 ). Then, for any n ≥ 1, any locally Lipschitz function f : Xn → [0, 1] and any s ∈ (0, 12 ), I (∫ Xn fdµn ) ≤ ∫ Xn √ I(f)2 + 4C2β2(s)|∇f |2dµn + CnsOsc(f). =-=(4)-=- Theorem 0.1 will easily follow from Theorem 0.2 by approximating indicator functions of sets by locally Lipschitz functions taking values in [0, 1]. Our starting point is the following one dimensiona... |

1 |
Sobolev spacesSpringer Series
- Maz′ja
- 1985
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Citation Context ...using the Young inequality xy ≤ Φ(2x) + Φ∗(y/2) with x = β(s)|∇f |/h and y = Φ′(|f |), where Φ∗(y) = supu{uy − Φ(u)}, we have β(s) ∫ X Φ′(|f |)|∇f |dµ ≤ ∫ X Φ (2β(s)|∇f |) dµ+ ∫ X Φ∗ (Φ′(|f |)/2) dµ. =-=(8)-=- A simple computation gives that Φ∗(y) = 1 − √ 1− y2 for |y| ≤ 1. Hence, since |Φ′| ≤ 1, we have Φ∗(Φ′(x)) = 1− √ 1− x21+x2 = √ 1+x2−1√ 1+x2 ≤ Φ(x) for any x ∈ R. In turn, using the convexity of Φ∗, w... |

1 | On the isoperimetric constants for product measures - Bobkov - 2009 |

1 | Problems in the theory of probability distributions - Steklov - 1974 |