The Localization Method for High-Dimensional Inequalities
This is an incremental survey that reviews an existing method with broad applications in areas like isoperimetric inequalities, optimization, and Markov chains, but does not introduce new results.
The paper surveys the localization method for proving high-dimensional inequalities, originally developed by Lovász and Simonovits and extended by Eldan, which transforms complex instances into simpler, often one-dimensional structured forms.
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.