Entropy Bounds and Isoperimetry

By S. G. Bobkov, B. Zegarlinski. 2005

95 citations recorded by the source corpus.

Subfields: Applied Mathematics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5f400107ae9def7e8eb68f15c127e7a3eed64149

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Citations in the corpus

Modified logarithmic Sobolev inequalities and transportation inequalities (2004, methodology): “Let us note that the cases α > 2 are studied by Bobkov-Ledoux in [BL00], relying mainly on Brunn-Minkowski inequalities, and by Bobkov-Zegarlinski in [BZ04] which refine the results presenting, via Hardy’s inequality, some necessary and sufficient condition for measures on the real line.”
Mass Transport and Variants of the Logarithmic Sobolev Inequality (2007, background): “[27] Bobkov S.G. and Zegarlinski B. Entropy bounds and isoperimetry.”
Isoperimetry between exponential and Gaussian (2006, background): “This result appears in [5, 18] in the particular case Φ(x) = |x|p.”
On the role of convexity in functional and isoperimetric inequalities (2008, methodology): “educes (up to constants) to Cheeger’s inequality, and the direction (1) ⇒ (2) to its reversed form due to Buser–Ledoux. In addition, using N(t) = tq log(1 + tq) and a result of Bobkov and Zegarlinski [18] (generalizing a previous result of Bobkov and Go¨tze [15]), a variant of Theorem 1.1 implies (s”
Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry (2010, result): “This result appears in [7; 24] in the particular case Φ(x) = |x|p and in [11] for Φ convex and √ Φ concave.”
Poincaré inequalities and dimension free concentration of measure (2010, methodology): “• Let p ≥ 2 and consider Hq(x) = |x|q with 1/p + 1/q = 1; the inequality LS(Hq,C) was studied by Bobkov and Ledoux in [13] and by Bobkov and Zegarlinski in [14], where a complete characterization on R was achieved (see [14], Theorem 5.”
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry (2004, background): “Recently, Bobkov and Zegarlinski [18] obtained concentration inequalities for these measures but for the ℓnα-distance on R n.”
Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary (2013, methodology): “r N<0. Remark9.3. As explained in Subsection 8.3, it is possible to prove the above assertions using the Borell–Brascamp–Lieb theorem. Another possibility is to invoke the localization method (see [28, 7]). However, these two approaches would be conﬁned to the Euclidean setting, whereas the proof”
Poincaré inequalities and hitting times (2010, background): “It is well known that one obtains a stronger inequality replacing the L 2 norm by a L p norm for 1 ≤ p ≤ 2 (see e.g. [7], Chapter 2).”
On the role of convexity in isoperimetry, spectral gap and concentration (2007, methodology): “is concludes the proof. Remark 5.12. The case q= 2 was previously shown by Ledoux [65]. For general 1 q2, the reverse direction without any convexity assumptions was shown by Bobkov and Zegarlinski [24], and given a dierent proof by Sodin and the author [74]. We will see a general argument for this ”
Coercive Inequalities on Metric Measure Spaces (2009, background): “More recently such criteria were established for other coercive inequalities (Log-Sobolev type: (LS2) [7] , (LSq) [10], for distributions with weaker tails [5],.”
A characterization of dimension free concentration in terms of transportation inequalities (2008, background): “Among the many functional inequalities yielding concentration estimates, let us mention the following: Poincaré inequalities [8, 23], logarithmic-Sobolev inequalities [7, 27], modified logarithmic-Sobolev inequalities [4, 8, 10, 18], transportation-cost inequalities [6, 7, 9, 20, 30, 33–35], inf-con”
Transport Inequalities. A Survey (2010, background): “Poincaré inequalities [55, 19], logarithmic Sobolev inequalities [67, 17], modified logarithmic Sobolev inequalities [19, 21, 47, 10], inf-convolution inequalities [82, 66], Beckner-Lata laOleszkiewicz inequalities [11, 65, 9, 7].”