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By using optimal mass transportation and a quantitative H\"older inequality,we provide estimates for the Borell-Brascamp-Lieb deficit on completeRiemannian manifolds. Accordingly, equality cases in Borell-Brascamp-Liebinequalities (including Brunn-Minkowski and Pr\'ekopa-Leindler inequalities)are characterized in terms of the optimal transport map between suitablemarginal probability measures. These results provide several qualitativeapplications both in the flat and non-flat frameworks. In particular, by usingCaffarelli's regularity result for the Monge-Amp\`ere equation, we {give a newproof} of Dubuc's characterization of the equality in Borell-Brascamp-Liebinequalities in the Euclidean setting. When the $n$-dimensional Riemannianmanifold has Ricci curvature ${\rm Ric}(M)\geq (n-1)k$ for some $k\in \mathbbR$, it turns out that equality in the Borell-Brascamp-Lieb inequality isexpected only when a particular region of the manifold between the marginalsupports has constant sectional curvature $k$. A precise characterization isprovided for the equality in the Lott-Sturm-Villani-type distortedBrunn-Minkowski inequality on Riemannian manifolds. Related results for (notnecessarily reversible) Finsler manifolds are also presented.

Equality in Borell-Brascamp-Lieb inequalities on curved spaces