# The Localization Method for High-Dimensional Inequalities

arXiv: https://arxiv.org/abs/2512.10848 · HTML: https://mlsanity.com/paper/2512.10848/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Dec 2025.

## Built on by (most cited first, 2 of 2)

- [Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run](https://mlsanity.com/paper/2608.16878/) (2026)
- [On the SoS Certifiability of Log-Concave Distributions](https://mlsanity.com/paper/2609.30105/) (2026)

## Abstract (arXiv)

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.