  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.
