8 problems

Slicing conjecture. The slicing constant is universally bounded:

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

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…

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…

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…

The hyperplane conjecture. There exists an absolute constant $C>0$ such that

Hyperplane conjecture. There exists a universal positive constant $c$ such that every convex set $K$ of unit volume in $\mathbb{R}^n$ has a hyperplane $H$ passing through its centr…