6 problems

Slicing conjecture. The slicing constant is universally bounded:

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…

For a convex body $K\subseteq\mathbb{R}^n$, let $L_K$ denote its isotropic constant. The isotropic constant conjecture. There is a universal constant $c>0$ such that, for every…

For $u\in\partial B_2^n$, define $u^\perp=\{x\in\mathbb{R}^n:\langle x,u\rangle=0\}$. For a convex body $K\subseteq\mathbb{R}^n$, let $b(K)=\int_K x\,dx/\lvert K\rvert$ be its bary…

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

Hyperplane conjecture. There exists a constant $C>0$ such that for every $n$ and every convex body $K\subset\mathbb{R}^n$ of volume $1$ and barycenter at the origin, there is a dir…