The Kannan-Lov\'asz-Simonovits Conjecture

TL;DR

The paper surveys the Kannan-Lovász-Simonovits conjecture, discussing its origins, implications across various fields, and recent progress in bounding the Cheeger constant for log-concave densities.

Contribution

It provides a comprehensive overview of the conjecture's background, significance, and the latest advances in bounding the Cheeger constant.

Findings

Current best bounds on the Cheeger constant for log-concave densities.

The conjecture's influence on techniques in geometry, probability, and algorithms.

Recent progress towards resolving the conjecture.

Abstract

The Kannan-Lov\'asz-Simonovits conjecture says that the Cheeger constant of any logconcave density is achieved to within a universal, dimension-independent constant factor by a hyperplane-induced subset. Here we survey the origin and consequences of the conjecture (in geometry, probability, information theory and algorithms) as well as recent progress resulting in the current best bounds. The conjecture has lead to several techniques of general interest.