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We prove an almost constant lower bound of the isoperimetric coefficient inthe KLS conjecture. The lower bound has the dimension dependency $d^{-o_d(1)}$.When the dimension is large enough, our lower bound is tighter than theprevious best bound which has the dimension dependency $d^{-1/4}$. Improvingthe current best lower bound of the isoperimetric coefficient in the KLSconjecture has many implications, including improvements of the current bestbounds in Bourgain's slicing conjecture and in the thin-shell conjecture,better concentration inequalities for Lipschitz functions of log-concavemeasures and better mixing time bounds for MCMC sampling algorithms onlog-concave measures.

An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture