Current Developments in Mathematics

Volume 2017

The Kannan–Lovász–Simonovits conjecture

Pages: 1 – 36



Yin Tat Lee (University of Washington, Seattle, Wa., U.S.A.)

Santosh S. Vempala (School of Mathematics, Georgia Institute of Technology, Georgia, U.S.A.)


The Kannan-Lovász–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.

This work was supported in part by NSF Awards CCF-1563838, CCF-1717349 and CCF-1740551.

Published 16 July 2019