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

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