### 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 language | English |
---|---|

Pages (from-to) | 453-494 |

Number of pages | 42 |

Journal | Advances in Mathematics |

Volume | 339 |

DOIs | |

Publication status | Published - Dec 1 2018 |

### Fingerprint

### 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

*Advances in Mathematics*,

*339*, 453-494. https://doi.org/10.1016/j.aim.2018.09.041

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

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 339, pp. 453-494. https://doi.org/10.1016/j.aim.2018.09.041

}

TY - JOUR

T1 - Equality in Borell–Brascamp–Lieb inequalities on curved spaces

AU - Balogh, Zoltán M.

AU - Kristály, A.

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

KW - Borell–Brascamp–Lieb inequality

KW - Brunn–Minkowski inequality

KW - Equality case

KW - Optimal mass transportation

KW - Prékopa–Leindler inequality

KW - Riemannian manifold

UR - http://www.scopus.com/inward/record.url?scp=85054312317&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85054312317&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2018.09.041

DO - 10.1016/j.aim.2018.09.041

M3 - Article

AN - SCOPUS:85054312317

VL - 339

SP - 453

EP - 494

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -