Balogh, Z.M. & Kristály, A. (2018) Advances in Mathematics [Matematică, Q1]

Publicat: 24 Noiembrie 2020

Balogh, Z.M. & Kristály, A. (2018) Equality in Borell–Brascamp–Lieb inequalities on curved spaces. Advances in Mathematics, 339, 453-494.

DOI: https://doi.org/10.1016/j.aim.2018.09.041

✓ Publisher: Elsevier
✓ Web of Science Core Collection: Science Citation Index Expanded
✓ Categories: Mathematics
✓ Article Influence Score (AIS): 1.976 (2018) / Q1

Abstract: By using optimal mass transportation and a quantitative Holder 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 Prekopa-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-Ampere 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 is an element of 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.

inapoi la stiri   vezi evenimentele   home