9 problems

Torquato–Stillinger conjecture. The function $g_2({\bf r})$ is a pair correlation function of a translationally invariant disordered sphere packing in $\mathbb{R}^d$ at number dens…

Exponential convergence conjecture. There exist absolute constants $c>0$ and $d_0\in\mathbb{N}$ such that, for all $d\geqslant d_0$,

Let $\mathcal V_{\text{typ}}$ be the typical cell of a Poisson–Voronoi tessellation in dimension $d$, with intensity $\lambda=\lambda(d)>0$, and let $B$ denote the unit ball in…

Small-cube conjecture. There is a unitary basis $L$ such that

Let $\Omega\subset\mathbb{R}^n$ be convex, bounded, and centrally symmetric, and let $E$ be an isoperimetric subset of $\Omega$ with $|E|/|\Omega|=1/2$. Two hyperplane conjecture.…

Let $\epsilon=\epsilon(n)$ be the radius of the geodesic balls, let $N_{c,\epsilon(n)}$ be the number of independently and uniformly selected centers used for the covering, and let…

Let $D_n$ be the Euclidean unit ball in $\mathbb{R}^n$, let $N$ be a positive integer, and let $P^b_{n,N}$ be a polytope with at most $N$ facets that is best-approximating for…

Let $f(t)=e^{-\alpha t^2}$, with $0<\alpha<\pi e$. For configurations of density $\rho$ in $\mathbb{R}^n$, let the minimal $f$-energy be the infimum of the corresponding Gaussian p…

Let $f$ be an isotropic down-monotone logconcave function in $\mathbb{R}^m$, and let $X$ have density $f$. Let $1>p>0$. The conditional-coordinate-tail conjecture. There exists an…