5 problems

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 … Write $\|\Sigma\|$…

Generalised variance conjecture. There is an absolute constant $C$ such that

Variance conjecture. There is an absolute constant, independent of $m$ and $K$, such that

Variance conjecture. The Poincaré inequality for the particular function $s$ above holds with a universal constant for every log-concave isotropic distribution $f$ in every dimensi…

For each $n$, define … Here isotropic means that $\mathbb{E}X=0$ and $\mathbb{E}(X\otimes X)=\operatorname{Id}_{\mathbb{R}^n}$. Variance conjecture. There exists a universal consta…