We study the Riemannian quantiative isoperimetric inequality. We show that direct analogue of the Euclidean quantitative isoperimetric inequality is--in general--false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) version of the quantitative isoperimetric inequality holds for a real analytic metric, using the Lojasiewicz-Simon inequality. A main novelty of our work is that in all our results we do not require any a priori knowledge on the structure shape of the minimizers.
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 ⋲ 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. For convex bodies D in Rn the deviation d from spherical shape is estimated from above in terms of the (dimensionless) isoperimetric deficiency A of D as follows: d < f(A) (for A sufficiently small). Here f is an explicit elementary function vanishing continuously at 0. The estimate is sharp as regards the order of magnitude of f . The dimensions n = 2 and 3 present anomalies as to the form of f . In the planar case n = 2 the result is contained in an inequality due to T. Bonnesen. A qualitative consequence of the present result is that there is stability in the classical isoperimetric problem for convex bodies D in Rn in the sense that, as D varies, d -p 0 for A -A 0. The proof of the estimate d < f(A) is based on a related estimate in the case of domains (not necessarily convex) that are supposed a priori to be nearly spherical in a certain sense.