The variance conjecture for symmetric convex bodies

Let $K\subset\mathbb{R}^d$ be a symmetric convex body, let $X$ be uniformly distributed on $K$, and let $\Sigma$ be the covariance matrix of $X$, with entries

$$\Sigma_{i,j}=\frac{1}{|K|}\int_K x_i x_j\,\mathrm{d}\boldsymbol{\mathbf{x}}.$$

Write $\|\Sigma\|$ for the operator norm and $\operatorname{tr}(\Sigma)$ for the trace. Variance conjecture. There is a constant $C\geq 1$ such that, for every dimension $d$ and every symmetric convex body $K\subset\mathbb{R}^d$,

$$\operatorname{Var}\|X\|_2^2\leq C\max_{\theta\in S^{d-1}}\mathbb{E}\langle X,\theta\rangle^2\cdot\mathbb{E}\|X\|_2^2=C\|\Sigma\|\operatorname{tr}(\Sigma).$$

This is the weaker form of the Kannan–Lovász–Simonovits conjecture obtained by taking $f(x)=\|x\|_2^2$. The paper studies the asymptotics of the associated inertia moments for Schatten balls; the conjecture remains unresolved in the generality stated here.