We prove an almost constant lower bound of the isoperimetric coefficient in the 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 the previous best bound which has the dimension dependency \(d^{-1/4}\). Improving the current best lower bound of the isoperimetric coefficient in the KLS conjecture has many implications, including improvements of the current best bounds in Bourgain’s slicing conjecture and in the thin-shell conjecture, better concentration inequalities for Lipschitz functions of log-concave measures and better mixing time bounds for MCMC sampling algorithms on log-concave measures.

我们证明了KLS猜想中等渗系数的几乎恒定的下限。下限具有尺寸依赖性\（d ^ {-o_d（1）} \）。当尺寸足够大时，我们的下限比具有尺寸依赖项\（d ^ {-1/4} \）的前一个最佳界限更紧密。改善KLS猜想的当前最佳等压系数下界具有许多含义，包括改进Bourgain切片猜想和薄壳猜想中的当前最佳界线，对数凹度度量的Lipschitz函数的更好的浓度不等式和更好的对数凹形度量中MCMC采样算法的混合时间范围。