Abstract
In this first chapter we introduce the two conjectures mentioned in the title of this monograph: The Kannan-Lovász-Simonovits conjecture, which was originally posed in relation with some problems in theoretical computer science, and the variance conjecture, which appeared independently in relation with the central limit problem for isotropic convex bodies and is a particular case of the KLS conjecture. The relation of the KLS conjecture with Cheeger’s isoperimetric inequality, as well as with Poincare’s inequality, the spectral gap of the Laplacian, and the concentration of measure phenomenon will be explained. Regarding the variance conjecture, it will be explained how this conjecture is equivalent to the thin-shell width conjecture and how it is implied by a strong property in some log-concave measures: The square negative correlation property.
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