Equality in Borell–Brascamp–Lieb inequalities on curved spaces

Zoltán M. Balogh, A. Kristály

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)453-494
Number of pages42
JournalAdvances in Mathematics
Volume339
DOIs
Publication statusPublished - Dec 1 2018

Fingerprint

Equality
Riemannian Manifold
Finsler Manifold
Optimal Transport
Ricci Curvature
Sectional Curvature
Probability Measure
n-dimensional
Euclidean
Regularity
Estimate

Keywords

  • Borell–Brascamp–Lieb inequality
  • Brunn–Minkowski inequality
  • Equality case
  • Optimal mass transportation
  • Prékopa–Leindler inequality
  • Riemannian manifold

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Equality in Borell–Brascamp–Lieb inequalities on curved spaces. / Balogh, Zoltán M.; Kristály, A.

In: Advances in Mathematics, Vol. 339, 01.12.2018, p. 453-494.

Research output: Contribution to journalArticle

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