Isoperimetry between exponential and Gaussian

By Franck Barthe, Patrick Cattiaux, Cyril Roberto. 2006

52 citations recorded by the source corpus.

We study the isoperimetric problem for 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

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/03d560d43c71c124434a44f2d1f693c22341c3d3

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

A generalization of Caffarelli’s contraction theorem via (reverse) heat flow (2010, unknown): “The tools to transfer these inequalities to the product measure have also recently been obtained [3, 4, 5, 47], and so consequently, the isoperimetric inequality satisfied by μ is well understood.”
Poincaré inequalities and dimension free concentration of measure (2010, methodology): “The following theorem due to Barthe, Cattiaux and Roberto shows how to deduce a super Poincaré inequality from a capacity measure inequality (see [6], Theorem 1 and Corollary 6).”
Lyapunov conditions for Super Poincaré inequalities (2009, background): “The use of this theorem requires some quite stringent conditions on η but covers the case η(u) = uα for 1 < α < 2, yielding a F -Sobolev inequality for F (u) = log+(u) (more general F are also studied in [5] section 7).”
Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré (2007, methodology): “First, since μ̄ also satisfies a Poincaré inequality on Id, we may apply Lemma 17 in [5] and obtain the following G-Sobolev inequality with G(u) = (log u − log 4)+ and some universal C (all universal constants will be denoted by C in the sequel) (5.”
On the role of convexity in functional and isoperimetric inequalities (2008, methodology): “In this sense the tensorization results of [6, 7] are sharp, but it would be interesting to obtain analogous results for arbitrary log-concave measures whose isoperimetric profile is not better than that of the Gaussian measure (due to the Central-Limit obstruction).”
Pointwise symmetrization inequalities for Sobolev functions and applications (2009, background): “[14]): there exists a universal constant c(Φ) such that IμΦ (t) inf n 1 IμΦ⊗n(t) c(Φ)IμΦ (t).”
Higher-order Sobolev embeddings and isoperimetric inequalities (2013, background): “The product probability spaces in R that are taken into account were analyzed in [3,4], and share common features with the Gauss space, namely R endowed with the probability measure dγn(x) = (2π)− n 2 e− |x|2 2 dx.”
Orlicz–Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups☆ (2006, background): “For other directions on the study of sub-Gaussian measures, the reader could like to see also [GGM05, BCR05a, BCR05c , Zeg01].”
Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry (2010, background): “For a large class of distributions μ on the line with exponential or faster decay, it is possible to prove [26; 52; 16; 20; 5; 10; 11; 47] that the isoperimetric profile Iμn of the n-tensor product μn is (up a to universal, hence dimension free constants) equal to Iμ.”
On the role of convexity in isoperimetry, spectral gap and concentration (2007, methodology): “ clear that an arbitrary tensor power cannot have an isoperimetric inequality better than the Gaussian measure . This explains the restriction on and above. In this sense the tensorization results of [9, 10] are sharp, but it would be interesting to obtain their results for arbitrary log-concave mea”