The general variance conjecture for centered log-concave vectors

Let $X$ be a centered log-concave random vector in $\mathbb R^n$, let $\lambda_X^2$ denote the largest eigenvalue of its covariance matrix, and let $|X|$ be its Euclidean norm. The general variance conjecture. There exists an absolute constant $C$ such that

$$\operatorname{Var}|X|^2\leq C\lambda_X^2\mathbb E|X|^2.$$

This is the non-isotropic specialization of the Kannan–Lovász–Simonovits conjecture with $g(X)=|X|^2$. The paper studies this conjecture and states that it is equivalent to a general thin-shell width conjecture.