5 problems

Bourgain's slicing conjecture. There exists a constant $c>0$ independent of dimension such that

Let $K\subseteq\mathbb{R}^n$ be centrally symmetric, let $K^\circ$ be its polar body, and let $(\vec{x},\vec{y})$ be uniformly distributed on $K\times K^\circ$. Polar second-moment…

Let $K\subseteq\mathbb{R}^n$ be a convex body, and let $L(K)$ be its isotropic constant. Isotropic constant conjecture. There is a constant $C$, independent of the dimension, such…

Let $f:\mathbb{R}^n\to\mathbb{R}_+$ be an integrable even log-concave function. With $f_1(x)=e^{-\sum_{i=1}^n|x_i|}$ and $f_\infty(x)=\chi_{[-1,1]^n}(x)$, even functional strong sl…

Let $L_f$, $\widetilde{L}_f$, and $\widehat{L}_f$ be the three functional isotropic constants defined for integrable centered log-concave functions. Define … Functional strong slic…