12 problems

Bourgain's hyperplane conjecture. There exists a universal constant $c>0$, independent of $n$, such that

Hyperplane conjecture. There is an absolute constant $C>0$ such that $L_K\leq C$ for every dimension $n$ and every $K\in\mathcal{K}_s^n$. The source recalls that this is equivalent…

Let $K\subseteq\mathbb R^n$ be a convex body, let $X$ be uniformly distributed on $K$, and define the isotropic constant of $K$ by … Let $\Delta^n$ be any simplex in $\mathbb R^n$.…

A convex body $K\subset\mathbb{R}^n$ is centrally symmetric if there exists a point $c\in\mathbb{R}^n$ such that $K-c=-(K-c)$. Let $C_n$ be an $n$-dimensional parallelepiped, and l…

Let $\mathcal{C}(K)=\lvert K\rvert^2/\det\operatorname{Cov}(K)=1/L_K^{2n}$ for a convex body $K\subseteq\mathbb{R}^n$. The strong isotropic constant conjecture for symmetric convex…

Let $\mathcal{C}(K)=\lvert K\rvert^2/\det\operatorname{Cov}(K)=1/L_K^{2n}$ for a convex body $K\subseteq\mathbb{R}^n$, and let $\Delta_{n,0}$ be the standard simplex. The strong is…

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…

Let $G\subset\mathbb{R}^d$ be a convex symmetric body in isotropic position, and let $L(G)$ denote its isotropic constant. Slicing conjecture. There is a constant $C>0$ independent…

The hyperplane conjecture. There is an absolute constant $C>0$, independent of the dimension $d$, such that

Kuperberg's conjecture. For every centrally symmetric convex body $K\subseteq\mathbb R^n$,

Slicing conjecture. The isotropic constants of symmetric convex bodies are bounded above by an absolute constant.

Let ${\mathcal IL}_{[n]}$ denote the class of isotropic log-concave probability measures on ${\mathbb R}^n$, let $I_{-q}(\mu)$ be the negative moment of the Euclidean norm, and for…