10 problems

Let $K\subset\mathbb R^n$ be an origin-symmetric convex body, let $GL_n$ denote the group of invertible linear transformations of $\mathbb R^n$, and let $\ell(K)$ be the product fu…

For each dimension $d$, let $L_d$ be the maximum of the isotropic constants $L_f$ among log-concave functions $f$ on $\mathbb{R}^d$. The isotropic constant conjecture. There exists…

Let $K$ be a convex body in isotropic position and of volume $1$ in $\mathbb{R}^d$, and let $\theta\in\mathbb{S}^{d-1}$ be a direction. The hyperplane conjecture. There exists a un…

Let $1le p le q < infty$ and, for each $nin\mathbb{N}$, let $\widetilde{X}^{(n)}=(\widetilde{X}^{(n)}_1,\dots,\widetilde{X}^{(n)}_n)$ be uniformly distributed on…

Let $K\subset\mathbb{R}^d$ be a symmetric convex body, let $X$ be uniformly distributed on $K$, and let $\Sigma$ be the covariance matrix of $X$, with entries … Write $\|\Sigma\|$…

Exponential growth conjecture. The ratio $R_{p,n}$ grows exponentially for every $0<p<\infty$.

Slicing conjecture. For any convex body $K$ in ${\mathbb R}^d$, there is a hyperplane $H$ such that the $(d-1)$-dimensional volume of the slice $H\cap K$ is bounded away from zero…

Let $B$ be an origin-symmetric convex body in ${\mathbb R}^n$ in the $\ell$-position. For $\varepsilon\in(0,1/2]$ and $k\leq c\varepsilon\log n/\log\frac{1}{\varepsilon}$, let…

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

Hyperplane conjecture. There exists an absolute constant $C$ such that, for every convex body $K$,