等周不等式的最优传输研究

Isoperimetric inequality is one of the most important geometric inequalities in mathematics, which has a long history and wide application. Gromov obtained a concise proof of isoperimetric inequality in 1986 by using optimal transport theory. Later Figalli-Maggi-Pratelli studied the stability of isoperimetric inequalities by using the optimal transport theory. Inspired by Figalli-Maggi-Pratelli's work, this paper uses Alzer's arithmetic-geometric mean inequality to optimize the parameters in the stability estimation of anisotropic isoperimetric inequality proved by Figalli-Maggi-Pratelli. The results are applied to Brunn-Minkowski inequality, and the parameters of stability estimation of Brunn-Minkowski inequality for convex collective product equality are optimized. Then, the method is applied to the stability estimation of Urysohn's inequality, and the stability estimation of Urysohn's inequality is obtained.