6 problems

Thin-shell conjecture. The sequence $(S_n)$ is bounded.

Let $X$ be an isotropic log-concave random vector in ${\mathbb R}^n$, and let $\sigma_4^2(X)$ denote the functional used in the source, which is equivalent to…

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

A log-concave random vector $X$ in $\mathbb R^n$ is isotropic if $\mathbb E X=0$ and its covariance matrix is the identity $I_n$. Write $\operatorname{Var}$ for variance and…

For each $n$, let $\sigma_n\geq0$ satisfy … where the supremum is over all isotropic log-concave random vectors $X$ in $\mathbb{R}^n$. Thin-shell conjecture. There exists a univers…

Anttila–Ball–Perissinaki thin-shell conjecture.