On the role of convexity in isoperimetry, spectral gap and concentration

By Emanuel Milman. 2007

269 citations recorded by the source corpus.

We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have arbitrarily slow uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov–Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “on-average” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan–Lovász–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semi-group following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. 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/c63b5ffd7451615fb9e3ea0f7a833695b1712a90

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Isoperimetric and Concentration Inequalities - Equivalence under Curvature Lower Bound (2009, methodology): “These assumptions are removed by an approximation argument detailed in Section 6. In Section 7 we show how Theorem 1.1 and the result of Section 3 may be used to give a direct proof of the Main Theorem from [ 34 ], and provide a proof of Theorem 1.3.”
Spectral gap for spherically symmetric log-concave probability measures, and beyond (2014, result): “A consequence of Bobkov’s theorem is that the class of spherically symmetric logconcave probability measures satisfies the KLS conjecture: if the covariance of µ is the identity, meaning that for all θ ∈ Rn, ∫ Rn |x · θ|2µ(dx) = ‖θ‖2, then the spectral gap λ1(−Lµ) is bounded from below by an univers”
Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds (2017, background): “It follows by a general result of the second-named author [49], which applies to any weighted-manifold satisfying the CD(0,∞) condition, and in particular to”
On the role of convexity in functional and isoperimetric inequalities (2008, methodology): “3] and the references therein), it was shown in [41] how to handle an arbitrary q 2.”
A generalization of Caffarelli’s contraction theorem via (reverse) heat flow (2010, unknown): “For instance, by [30] (see also [48]), one may estimate λ1 ≥ c( ∫ |x|dμ(x))−2 > 0 for some universal numeric constant c > 0.”
Pointwise symmetrization inequalities for Sobolev functions and applications (2009, methodology): “Milman [96]), we know that I (t)/t is decreasing and I (t) is symmetric about 1/2 so by a convexity argument we can deduce that I (t) t, t ∈ (0,1/2] concluding the proof.”
Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures (2010, methodology): “Let us finally mention that by a standard application of a remarkable theorem due to Bobkov [6], we improve the best-known general bound on the Cheeger constant DChe(μ) of a probability measure μ in Rn with isotropic log-concave density (we refer to [6, 29] for missing definitions and background).”
Thin Shell Implies Spectral Gap Up to Polylog via a Stochastic Localization Scheme (2012, background): “For an extensive review of this constant and equivalent formulations, see [Mil1].”
Logarithmic bounds for isoperimetry and slices of convex sets (2023, background): “Milman’s theorem [37] states that cλ 0 ≤ sup φ is 1-Lipschitz V arμ(φ) (30) where the supremum runs over all 1-Lipschitz functions φ : R → R, and where c > 0 is a universal constant.”
Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order (2013, background): “[52]) that if X satisfies the Poincaré inequality Var ( f (X)) ≤ DPoinE|∇ f (X)|2 (14) for all locally Lipschitz bounded functions, then X satisfies (11) with γ = 1 and L = C√DPoin (recall that C always denotes a universal constant).”
Sharp convex Lorentz–Sobolev inequalities (2011, unknown): “…ya [56] deduced (2) from the Maz (cid:48) ya L p isocapacitary inequality (see [56]), (3) , and the Maz (cid:48) ya capacity strong-type inequality [53], where C p ( K ) is the L p variational capacity of K deﬁned by For more connections of (3) to Sobolev-type embeddings, see [26,55,57–60,62,76].”
Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary (2013, background): “ satisﬁes the CD(0,N) condition with N∈ [n,∞] and II∂M ≥ 0 (M is locally convex), it is known that v→ I(v) N N−1 is in fact concave on its domain, implying that v→ I(v) N N−1 /vis non-increasing (see [41, 42] and the references therein). The proof of this involves crucial use of regularity results f”
An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture (2020, methodology): “First, according to the concavity of the isoperimetric profile [16] (Theorem 1.”
Transport Inequalities. A Survey (2010, unknown): “In a series of papers [87, 86, 85], E.”
Transport Inequalities for Log-concave Measures, Quantitative Forms, and Applications (2015, unknown): “10) c h(μ)2 ≤ λ(μ) ≤ C h(μ)2 for some universal (numerical) constants c,C > 0, independent of μ and n; see [22, 25].”