9 problems

Let $S\subset\mathbb{S}^{n-1}$ and let $\delta>0$. A matrix $G\in\mathbb{R}^{m\times n}$ induces a $\delta$-uniform tessellation of $S$ if, for all $x,y\in S$, … where…

Let $\Delta_n$ be the regular simplex of side length $\sqrt{2}$, and let $H$ range over hyperplanes passing through the center of $\Delta_n$. Write $H_{\text{facet}}$ for any hyper…

Gaussian tessellation embedding conjecture. For random homogeneous Gaussian tessellations of $T$, the condition

Milman-moduli conjecture. For positive $\varepsilon$,

Day–Nordlander conjecture. The displayed inequalities hold.

Hyperplane conjecture. There exists an absolute constant $c$ such that every convex body of volume $1$ has a hyperplane section of volume greater than $c$.

Let $K\subseteq S^{n-1}$, let $m(K,\delta)$ be the smallest number of hyperplanes that provide a $\delta$-uniform tessellation of $K$, and let $w(K)$ denote the mean width of $K$.…

Let ${\bf K}$ be an arbitrary $0$-symmetric convex body in ${\mathbb R}^3$. A truncated octahedron of the form ${\bf T}-{\bf T}$ is obtained from an arbitrary tetrahedron ${\bf T}$…

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…