Equality in Borell-Brascamp-Lieb inequalities on curved spaces

Balogh, Zoltan; Kristaly, Alexandru (2018). Equality in Borell-Brascamp-Lieb inequalities on curved spaces. Advances in mathematics, 339, pp. 453-494. Elsevier 10.1016/j.aim.2018.09.041

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By using optimal mass transportation and a quantitative Hölder inequality, we provide estimates for the Borell–Brascamp–Lieb deficit on complete Riemannian manifolds. Accordingly, equality cases in Borell–Brascamp–Lieb inequalities (including Brunn–Minkowski and Prékopa–Leindler inequalities) are characterized in terms of the optimal transport map between suitable marginal probability measures. These results provide several qualitative applications both in the flat and non-flat frameworks. In particular, by using Caffarelli's regularity result for the Monge–Ampère equation, we give a new proof of Dubuc's characterization of the equality in Borell–Brascamp–Lieb inequalities in the Euclidean setting. When the n-dimensional Riemannian manifold has Ricci curvature Ric(M) ≥ (n-1)k for some k ⋲ ℝ, it turns out that equality in the Borell–Brascamp–Lieb inequality is expected only when a particular region of the manifold between the marginal supports has constant sectional curvature k. A precise characterization is provided for the equality in the Lott–Sturm–Villani-type distorted Brunn–Minkowski inequality on Riemannian manifolds. Related results for (not necessarily reversible) Finsler manifolds are also presented.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Balogh, Zoltan and Kristaly, Alexandru

Subjects:

500 Science > 510 Mathematics

ISSN:

0001-8708

Publisher:

Elsevier

Language:

English

Submitter:

Olivier Bernard Mila

Date Deposited:

22 May 2019 14:23

Last Modified:

24 Oct 2019 08:01

Publisher DOI:

10.1016/j.aim.2018.09.041

BORIS DOI:

10.7892/boris.125485

URI:

https://boris.unibe.ch/id/eprint/125485

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