论文标题
具有同质重量的同余不平等的急剧定量稳定性
Sharp quantitative stability for isoperimetric inequalities with homogeneous weights
论文作者
论文摘要
我们证明了广泛的加权等值不平等的急剧定量稳定性。更确切地说，我们考虑具有均匀重量的凸锥中的等速度不平等。 受ABP方法的这种等值不平等的证明的启发，我们构建了一个新的凸耦合（即，在通用集合$ e $和不平等的最小化器（如Gromov证明了Gromov的Isoperimetric不等式不等式的证明）之间，我们构建了一个新的凸耦合（即是凸功能的梯度）。即使该地图不是来自最佳传输，即使不平等的权重，我们也适应了Figalli-Maggi-Pratelli的方法，并证明，如果$ E $对于不平等的几乎是最佳的，那么它在定量上就可以在最小化的最小化合物中进行最小化。然后，必须进行微妙的分析以排除翻译的可能性。 作为我们证明的一步，我们为可能具有独立关注的功能的凸凸包络建立了鲜明的规律性结果。
We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequalities through the ABP method, we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set $E$ and the minimizer of the inequality (as in Gromov's proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of Figalli-Maggi-Pratelli and prove that if $E$ is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations. As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest.
