The Localization Method for High-Dimensional Inequalities
Probability
2026-03-25 v2 Data Structures and Algorithms
Functional Analysis
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.
Cite
@article{arxiv.2512.10848,
title = {The Localization Method for High-Dimensional Inequalities},
author = {Yunbum Kook and Santosh S. Vempala},
journal= {arXiv preprint arXiv:2512.10848},
year = {2026}
}
Comments
v2: Add more details on the classical method. Include the thin-shell proof in arXiv:2507.15495