The KLS constant is $O({\log }^{1/4}n)$

Abstract

We confirm the Kannan--Lovász--Simonovits conjecture for quadratic forms: if $X\sim \hat{I}¼$ is an isotropic log-concave random vector in ${\mathbb{R}}^n$, then for any symmetric matrix $M$ one has $${\mathrm{Var}}_{X\sim \hat{I}¼}(⟨MX,X⟩)\le 2{\mathbb{E}}_{X\sim \hat{I}¼}|\nabla ⟨MX,X⟩{|}^2.$$ As an application, we apply the above to $M={\mathbb{E}}_{X\sim \hat{I}¼}(⟨X,\hat{I}¸⟩X\otimes X)$ for $\hat{I}¸\in S^{n-1}$ and show that the Kannan--Lovász--Simonovits constant $\ddot{I}{ˆ}_n$ satisfies $$\ddot{I}{ˆ}_n\le C{\log }^{1/4}n$$ for some absolute constant $C>0$.

21 pages