We survey the localization method for proving inequalities in high dimension, pioneered by Lovász and Simonovits (1993), and its stochastic extension developed by Eldan (2012). The method has found applications in a surprising wide variety of settings, ranging from its original motivation in isoperimetric inequalities to optimization, concentration of measure, and bounding the mixing rate of Markov chains. At heart, the method converts a given instance of an inequality (for a set or distribution in high dimension) into a highly structured instance, often just one-dimensional.

翻译：本文综述了由Lovász与Simonovits（1993）开创的用于证明高维不等式的局部化方法，及其由Eldan（2012）发展的随机扩展。该方法已在令人惊讶的广泛领域中得到应用，从最初源于等周不等式的动机，延伸至优化、测度集中以及马尔可夫链混合速率的界估计。其核心思想是将给定不等式（针对高维空间中的集合或分布）的一个实例转化为高度结构化的实例，通常仅为一维情形。