25 problems

A sphere packing is a collection of congruent non-overlapping spheres in Euclidean space; it is disordered when it lacks the long-range order characteristic of a lattice packing. T…

A lattice $\Lambda\subset\mathbb{R}^n$ is a discrete subgroup of rank $n$, and its packing density is the density of the packing by spheres centered at lattice points with radius h…

Consider a multiple-description lattice vector quantization system with $K$ descriptions, and let the side distortions be evaluated in the high-resolution regime as the lattice dim…

Let $K$ be the number of descriptions, let $L$ be the lattice dimension, and let $N$ and $us$ be the system parameters in the paper. For each pair of distinct description indices…

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

A lattice packing in Euclidean space is a sphere packing whose centers form a lattice, and a lattice packing is suboptimal when its density is strictly less than the optimal densit…

Let $N$ points in the relevant $d$-dimensional domain be chosen deterministically, and let the generalized discrepancy be measured in the $p=1$ case. Curse-of-dimensionality conjec…

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…

Consider the minors that are forbidden for embeddability of high-dimensional manifolds in the relevant Euclidean spaces. Exponential forbidden-minor conjecture. As the dimension in…

Let $\bigcirc^d$ denote the Euclidean unit ball in dimension $d$, and let $P_{\bigcirc^d}^{(d)}(n)$ be the probability that $n$ independent uniformly random points in this ball are…

Let $d\geq 3$, let $K\subset\mathbb{R}^d$ be a convex domain of volume $1$, and let $\triangle^d$ and $\bigcirc^d$ denote respectively a simplex and an ellipsoid of volume $1$. Wri…

Let ${\cal H}_n:= \{-1,+1\}^n$ be the bipolar cube, and call two vectors $\varepsilon$-almost orthogonal if the angle between them differs from $\pi/2$ by at most $\varepsilon$. An…

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

Zero-weight discrepancy conjecture. The improvement of the discrepancy estimate for weights equal to zero in the neighborhood of $t=n/2$ should also take place for $d$-dimensional…

Let $N(\varepsilon,d)$ denote the least number of points in the $d$-dimensional unit cube needed to obtain dispersion at most $\varepsilon$. Ullrich–Vybíral conjecture. The quantit…

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 $c\in\mathbb R^{+}$. For sufficiently large dimension and sufficiently small $\epsilon$, define … Here $N_{c,\epsilon(n)}$ is the prescribed number of geodesic balls and…

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 $N(\epsilon,n)$ be the number of geodesic balls considered in the sphere-covering problem, and let $c_1>0$ denote an absolute constant independent of $\epsilon$ and $n$. The ex…

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 $\mathcal{B}_2$ be the unit $\ell_2$-ball centered at the origin in $\mathbb{R}^n$, and let $\lambda$ be fixed. The quantity $\#(\lambda\mathcal{B}_2\cap\mathbb{Z}^n)$ counts t…

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…

Maximum-threshold conjecture. The threshold among all such systems is maximized by hyperspheres:

Universality Hypothesis. The probability of unique solution under the Gaussian Ensemble is the same as the probability of unique solution under each other ensemble in the universal…

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…