Uniform tail-decay of Lipschitz functions implies Cheeger's isoperimetric inequality under convexity assumptions

Milman, Emanuel

Comptes Rendus. Mathématique, Tome 346 (2008) no. 17-18, pp. 989-994.

Nous montrons que pour les domaines convexes dans l'espace euclidien, l'inégalité isopérimétrique de Cheeger, l'existence du trou spectral pour le Laplacien de Neumann, la concentration exponentielle des fonctions lipschitziennes et la a priori plus faible propriété de queue-affaiblissement uniforme de ces fonctions, sont toutes équivalentes (à constantes universelles près, indépendamment de la dimension). Ceci étend considérablement des résultats précédents de Maz'ya, Cheeger, Gromov–Milman, Buser et Ledoux. Comme application, nous en déduisons la stabilité du trou spectral des domaines convexes sous perturbations convexes qui préservent le volume (à des constantes près). Nous offrons aussi une nouvelle caractérisation de la constante de Cheeger, comme l'inverse de la moyenne de la distance par rapport au « pire » ensemble borélien ayant la moitié de la mesure du domaine convexe. En outre, nous récupérons facilement (et prolongez) beaucoup de limites inférieures précédemment connues dues à Payne–Weinberger, Li–Yau et Kannan–Lovász–Simonovits, sur la constante de Cheeger des domaines convexes. Nos résultats s'étendent plus généralement aux variétés riemanniennes munies d'une densité qui satisfont la condition de courbure-dimension CD(0,∞) de Bakry–Émery.

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 uniform tail-decay of these functions, are all 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 the stability of the spectral-gap for 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 of the Cheeger constant, as one over the expectation of the distance from the “worst” Borel set having half the measure of the convex domain. In addition, we easily recover (and extend) many previously known lower bounds, due to Payne–Weinberger, Li–Yau and Kannan–Lovász–Simonovits, on the Cheeger constant of convex domains. Essential to our proof is 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.

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