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

Abstract

We confirm the Kannan--Lov\'asz--Simonovits conjecture for quadratic forms: if $X \sim \mu$ is an isotropic log-concave random vector in $\mathbb{R}^n$, then for any symmetric matrix $M$ one has $$ \operatorname{Var}_{X \sim \mu}(\langle MX,X\rangle) \leq 2\,\mathbb{E}_{X \sim \mu}|\nabla\langle MX,X\rangle|^2. $$ As an application, we apply the above to $M=\mathbb{E}_{X \sim \mu}(\langle X,\theta\rangle X\otimes X)$ for $\theta\in S^{n-1}$ and show that the Kannan--Lov\'asz--Simonovits constant $\psi_n$ satisfies $$ \psi_n\leq C\log^{1/4}n $$ for some absolute constant $C>0$.