Simplified proof of Klartag's CLT for convex bodies via log-concave functions, with appendix on thin shell implying CLT.
The Kannan-Lov\'asz-Simonovits Conjecture
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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.
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2019 1verdicts
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A simplified proof of CLT for convex bodies
Simplified proof of Klartag's CLT for convex bodies via log-concave functions, with appendix on thin shell implying CLT.