The Localization Method for High-Dimensional Inequalities
Abstract
We survey the localization method for proving inequalities in high dimension, pioneered by Lov\'asz 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.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.