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.
The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. In this paper we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman. More generally, the result applies to every optimal Poincare-Sobolev constant for the embeddings @math . A version of the Riesz-Sobolev convolution inequality is formulated and proved for arbitrary compact connected Abelian groups. Maximizers are characterized and a quantitative stability theorem is proved, under natural hypotheses. A corresponding stability theorem for sets whose sumset has nearly minimal measure is also proved, sharpening recent results of other authors. For the special case of the group @math , a continuous deformation of sets is developed, under which an appropriately scaled Riesz-Sobolev functional is shown to be nondecreasing. In this note, we establish a strong form of the quantitive Sobolev inequality in Euclidean space for @math . Given any function @math , the gap in the Sobolev inequality controls @math , where @math is an extremal function for the Sobolev inequality. Abstract We prove a quantitative stability result for the Brunn–Minkowski inequality: if | A | = | B | = 1 , t ∈ [ τ , 1 − τ ] with τ > 0 , and | t A + ( 1 − t ) B | 1 n ≤ 1 + δ for some small δ , then, up to a translation, both A and B are quantitatively close (in terms of δ ) to a convex set K .   We present some recent stability results concerning the isoperimetric inequality and other related geometric and functional inequalities. The main techniques and approaches to this field are discussed.