Harmonic Measures on the Sphere via Curvature-Dimension

Milman, Emanuel

doi:10.5802/afst.1540

Cited by 10 publications

(7 citation statements)

References 32 publications

(39 reference statements)

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“…[44,27]); an exception is the work of Ohta and Takatsu [40,41]. We expect this gap in the literature to be quickly filled (in fact, concurrently to posting our work on the arXiv, Ohta [39] has posted a first attempt of a systematic treatise of the range N ≤ 0, and subsequently other authors have also begun treating this extended range [32,15,50,14,33]). A convenient equivalent form of the CD(ρ, N ) condition may be formulated as follows.…”

Section: Introductionmentioning

confidence: 97%

Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary

Kolesnikov

Milman

2016

J Geom Anal

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It is known that by dualizing the Bochner-Lichnerowicz-Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). When the manifold has a boundary, an appropriate generalization of the Reilly formula may be used instead. By systematically dualizing this formula for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet), we obtain new Brascamp-Lieb type inequalities on the manifold. All previously known inequalities of Lichnerowicz, Brascamp-Lieb, Bobkov-Ledoux and Veysseire are recovered, extended to the Riemannian setting and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are obtained. Our framework allows to encompass the entire class of Borell's convex measures, including heavy-tailed measures, and extends the latter class to weighted-manifolds having negative generalized dimension.

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Section: Introductionmentioning

confidence: 97%

Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary

Kolesnikov

Milman

2016

J Geom Anal

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“…In the Riemannian case, Ric N with N ∈ [n, ∞] has been well studied, see [Li, Ba, Qi] among many others. The study of the negative range N ∈ (−∞, 0) (and even N ∈ (−∞, 1]) is more recent, see [KM,Mi1,Mi2,Oh5,Wy,WY].…”

Section: Weighted Ricci Curvaturementioning

confidence: 99%

Some functional inequalities on non-reversible Finsler manifolds

Ohta

2017

Proc Math Sci

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We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the Γ-calculusà la Bakry et al, we show the dimensional versions of the Poincaré-Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti-Mondino in the framework of curvature-dimension condition by means of the localization method. We show that the same (sharp) estimates hold also for non-reversible metrics.

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“…Klartag's work [Kl] also covers N ∈ (−∞, 1) ∪ [n, ∞]. See [Mi3] for a recent interesting example equipped with N ∈ (−∞, n).…”

Section: Weighted Ricci Curvaturementioning

confidence: 99%

Needle decompositions and isoperimetric inequalities in Finsler geometry

Ohta

2018

J. Math. Soc. Japan

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Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Lévy-Gromov, Bakry-Ledoux, Bayle and E. Milman on weighted Riemannian manifolds. Klartag's approach is based on a generalization of the localization method (so-called needle decompositions) in convex geometry, inspired also by optimal transport theory. Cavalletti and Mondino subsequently generalized the localization method, in a different way more directly along optimal transport theory, to essentially non-branching metric measure spaces satisfying the curvaturedimension condition. This class in particular includes reversible (absolutely homogeneous) Finsler manifolds. In this paper, we construct needle decompositions of non-reversible (only positively homogeneous) Finsler manifolds, and show an isoperimetric inequality under bounded reversibility constants. A discussion on the curvature-dimension condition CD(K, N ) for N = 0 is also included, it would be of independent interest.Mathematics Subject Classication (2010): 53C60, 49Q20

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Harmonic Measures on the Sphere via Curvature-Dimension

Cited by 10 publications

References 32 publications

Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary

Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary

Some functional inequalities on non-reversible Finsler manifolds

Needle decompositions and isoperimetric inequalities in Finsler geometry

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