# Equality in Borell-Brascamp-Lieb inequalities on curved spaces

## TL;DR

This paper characterizes equality cases in Borell-Brascamp-Lieb inequalities on curved spaces using optimal transport, providing new proofs and insights into geometric and curvature-related conditions for equality.

## Contribution

It introduces a novel approach combining optimal mass transportation and a quantitative H"older inequality to analyze equality cases in Borell-Brascamp-Lieb inequalities on Riemannian and Finsler manifolds.

## Key findings

- Characterization of equality cases via optimal transport maps.
- New proof of Dubuc's Euclidean equality characterization.
- Conditions for equality related to constant sectional curvature regions.

## Abstract

By using optimal mass transportation and a quantitative H\"older 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 Pr\'ekopa-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-Amp\`ere 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 ${\rm Ric}(M)\geq (n-1)k$ for some $k\in \mathbb R$, 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.