The Localization Method for High-Dimensional Inequalities

TL;DR

This paper reviews the localization method for proving high-dimensional inequalities, highlighting its development, applications, and how it reduces complex problems to simpler, often one-dimensional, instances.

Contribution

It provides a comprehensive survey of the localization method and its stochastic extension, emphasizing its versatility across various mathematical and probabilistic fields.

Findings

The localization method simplifies high-dimensional inequalities to one-dimensional problems.

It has broad applications in isoperimetric inequalities, optimization, and Markov chain analysis.

The stochastic extension enhances the method's applicability in probabilistic settings.

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.