Gaussian-type isoperimetric inequalities in RCD $(K, \infty)$ probability spaces for positive $K$

Ambrosio, Luigi; Mondino, Andrea

doi:10.4171/rlm/745

Cited by 12 publications

(9 citation statements)

References 24 publications

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“…In summary, we get Appying Proposition 4.1, we obtain the functional version of Gaussian isoperimetric inequality of Bobkov on RCD(K, ∞) spaces, which had been proved by Ambrosio-Mondino in [11] using a different proof (see also [15,Chapter 8.5.2] for more discussions). Proposition 4.2.…”

Section: Rigidity Of Some Functional Inequalitiesmentioning

confidence: 60%

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Rigidity of some functional inequalities on RCD spaces

Han¹

2020

Preprint

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We study the cases of equality and prove a rigidity theorem concerning the 1-Bakry-Émery inequality. As an application, we prove the rigidity and identify the extremal functions of the Gaussian isoperimetric inequality, the logarithmic Sobolev inequality and the Poincaré inequality in the setting of RCD(K, ∞) metric measure spaces. This unifies and extends to the non-smooth setting the results of Carlen-Kerce [20], Morgan [43], Bouyrie [19], Ohta-Takatsu [44], Cheng-Zhou [23].Examples of non-smooth spaces fitting our setting are measured-Gromov Hausdorff limits of Riemannian manifolds with uniform Ricci curvature lower bound, and Alexandrov spaces with curvature lower bound. Some results including the rigidity of the 1-Bakry-Émery inequality, the rigidity of Φ-entropy inequalities are of particular interest even in the smooth setting.

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Section: Rigidity Of Some Functional Inequalitiesmentioning

confidence: 60%

“…In [16], Bakry and Ledoux proved the Bobkov's inequality (1.7) on smooth metric measure spaces using a semigroup method. Recently, by adopting the argument of Bakry-Ledoux, Ambrosio-Mondino [11] obtain the Bobkov's inequality in the non-smooth RCD(K, ∞) setting.…”

Section: Introductionmentioning

confidence: 99%

Rigidity of some functional inequalities on RCD spaces

Han¹

2020

Preprint

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“…First of all, the best results obtained before this paper are the aforementioned estimates (6)-(7) due to Buser [12] and Ledoux [24] for smooth complete Riemannian manifolds satisfying Ric ≥ K , K ≤ 0. Let us stress that the constants in Corollary 1.2 improve the ones in both (6)- (7) and are dimension-free as well. In addition, the improvements of the present paper are:…”

Section: Comparison With Previous Results In the Literaturementioning

confidence: 90%

“…Finally various lower bounds, together with rigidity and almost rigidity statements for the Dirichlet first eigenvalue of the Laplacian, have been proved by Mondino-Semola [ 29 ] in the framework of and spaces. Lower bounds on Cheeger’s isoperimetric constant have been obtained for (essentially non-branching) spaces by Cavalletti-Mondino [ 13 – 15 ] and for spaces ( ) by Ambrosio-Mondino [ 7 ]. The local and global stability properties of eigenvalues and eigenfunctions in the framework of spaces have been investigated by Gigli-Mondino-Savaré in [ 20 ], and by Ambrosio-Honda in [ 2 , 3 ].…”

Section: Introductionmentioning

confidence: 99%

Sharp Cheeger–Buser Type Inequalities in $$ \mathsf {RCD}(K,\infty )$$ Spaces

Ponti

Mondino

2020

J Geom Anal

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The goal of the paper is to sharpen and generalise bounds involving Cheeger's isoperimetric constant h and the first eigenvalue λ 1 of the Laplacian. A celebrated lower bound of λ 1 in terms of h, λ 1 ≥ h 2 /4, was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on λ 1 in terms of h was established by Buser in 1982 (with dimensional constants) and improved (to a dimension-free estimate) by Ledoux in 2004 for smooth Riemannian manifolds with Ricci curvature bounded below. The goal of the paper is twofold. First: we sharpen the inequalities obtained by Buser and Ledoux obtaining a dimension-free sharp Buser inequality for spaces with (Bakry-Émery weighted) Ricci curvature bounded below by K ∈ R (the inequality is sharp for K > 0 as equality is obtained on the Gaussian space). Second: all of our results hold in the higher generality of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below in synthetic sense, the so-called RCD(K , ∞) spaces.

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“…By Proposition 2.14 and assumption (10) there exists a Borel set E ⊂ X with 0 < m(E) ≤ 1/2 such that Per(E)…”

Section: Proof Of Theorem 12mentioning

confidence: 98%

The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces

De Ponti,

Mondino,

Semola

2020

Preprint

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We prove that the sharp Buser's inequality obtained in the framework of RCD(1, ∞) spaces by the first two authors [26] is rigid, i.e. equality is obtained if and only if the space splits isomorphically a Gaussian. The result is new even in the smooth setting. We also show that the equality in Cheeger's inequality is never attained in the setting of RCD(K, ∞) spaces with finite diameter or positive curvature, and we provide several examples of spaces with Ricci curvature bounded below where these assumptions are not satisfied and the equality is attained. As a consequence of the two main results, we obtain improved versions of Buser's and Cheeger's inequalities for RCD(K, N ) spaces which are new even for smooth Riemannian manifolds of dimension N with Ricci curvature bounded below by K ∈ R.

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Gaussian-type isoperimetric inequalities in RCD $(K, \infty)$ probability spaces for positive $K$

Cited by 12 publications

References 24 publications

Rigidity of some functional inequalities on RCD spaces

Rigidity of some functional inequalities on RCD spaces

Sharp Cheeger–Buser Type Inequalities in $$ \mathsf {RCD}(K,\infty )$$ Spaces

The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces

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