On the role of convexity in functional and isoperimetric inequalities

By Emanuel Milman. 2008

55 citations recorded by the source corpus.

This is a continuation of our previous work [Preprint, 2008, http://arxiv.org/abs/0712.4092]. It is well known that various isoperimetric inequalities imply their functional ‘counterparts’, but in general this is not an equivalence. We show that under certain convexity assumptions (for example, for log‐concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz–Sobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz'ya and extended by Barthe–Cattiaux–Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central‐Limit obstruction. As another application, we show that under our convexity assumptions, q‐log‐Sobolev inequalities (q ∈ [1, 2]) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry–Ledoux and Bobkov–Zegarliński. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvature–dimension condition of Bakry–Émery.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b16b592bd088b5053debdfba646b06b7f51d8926

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Transport Inequalities. A Survey (2010, unknown): “In a series of papers [87, 86, 85], E.”
Higher-order Sobolev embeddings and isoperimetric inequalities (2013, background): “The contributions to this field now constitute the corpus of a vast literature, which includes the papers [1, 4, 6, 10, 11, 15, 16, 20, 22, 24, 27, 31, 33, 39, 40, 45, 47, 49, 50, 52, 53, 54, 59, 60, 68, 89] and the monographs [17, 19, 21, 48, 67, 78].”
Isoperimetric and Concentration Inequalities - Equivalence under Curvature Lower Bound (2009, background): “ theorem. We emphasize that it is not true in general that if {µm} tends to µin total variation on a common metric space (Ω,d), then necessarily I(Ω,d,µ m) tends to I(Ω,d,µ), even pointwise (see e.g. [34, 33]). Fortunately, a one-sided relation always exists. Recall that {µm} is said to converge to ”
Pointwise symmetrization inequalities for Sobolev functions and applications (2009, methodology): “Estimating isoperimetric profiles via semigroups In this section we discuss an extension of the approach in [96,97] to the self improving results in Section 6.”
Sharp Isoperimetric Inequalities and Model Spaces for Curvature-Dimension-Diameter Condition (2011, background): “Federer–Fleming [31], Maz’ya [54] and Cheeger [25]) that isoperimetric inequalities imply corresponding Sobolev inequalities, and it is also known (see Ledoux [47] and also [56]) that Sobolev inequalities may be strengthened to isoperimetric inequalities under a curvature lower bound.”
On the role of convexity in isoperimetry, spectral gap and concentration (2007, background): “We provide the argument in [64] and omit it here, since it is not required for the results of this work.”
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.”