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和Simonovits（1993）开创的高维不等式证明局部化方法，以及Eldan（2012）发展的随机扩展版本。该方法已在令人惊讶的广泛领域中取得应用，从最初等周不等式的动机出发，延伸至优化理论、测度集中现象以及马尔可夫链混合速率的界估计。该方法的核心思想在于将高维空间中的集合或分布所满足的不等式实例，转化为高度结构化的实例（通常仅为一维情形）。