The Localization Method for High-Dimensional Inequalities
By: Yunbum Kook, Santosh S. Vempala
Potential Business Impact:
Simplifies hard math problems into easy ones.
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.
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