17 problems

Minkowski-type criterion for three-dimensional lattice coverings. The arrangement $K+\mathbb{Z}^3$ is a lattice covering of $\mathbb{R}^3$ if and only if there exist a unimodular t…

Let $B_n$ be the closed Euclidean unit ball in ${\bf R}^n$, let $K_n$ be the unit cube in ${\bf R}^n$, and let $\beta(B_n,K_n)$ denote the corresponding lattice covering quantity.…

Three-dimensional covering-density conjecture.

Three-dimensional lattice covering conjecture. The lattice coverings obtained from the authors' Cayley-graph constructions are optimal.

Let $\mathsf{E}_8$ denote the root lattice in dimension $8$. A globally optimal solution to the lattice packing-covering problem is one attaining the best packing-covering value am…

Let $\mathsf{E}_8$ denote the root lattice in dimension $8$. A lattice is a locally optimal solution to the lattice packing-covering problem if no sufficiently small deformation im…

Let $Q^{c1}_6$ be the positive quadratic form defining the lattice covering studied in the paper, and let lattice coverings in dimension $6$ be compared by their covering density.…

A lattice in dimension $6$ determines a positive definite quadratic form and has a covering density, namely the density of congruent balls centered at lattice points needed to cove…

Algorithmic verification criterion. The following conditions are equivalent: the configuration $\lambda K+\mathbb{Z}^3$ is a lattice covering realizing the specified combinatorial…

Algebraic intersection criterion. The arrangement $\lambda K+\mathbb{Z}^3$ is a lattice covering of $\mathbb{R}^3$ if and only if there exists a unimodular transformation…

Boundary-vertex conjecture. The vertices of $Q$ lie on the boundary of $K-\mathbf{x}$.

Symmetric minimal covering body classification conjecture. There exists a unimodular transformation $T$ such that

Lattice covering density conjecture. For all $K\in\mathcal{K}^{n}$, there is an absolute constant $c>0$ such that

Let $T$ be a regular tetrahedron in three-dimensional space, and let $\vartheta_L(T)$ denote its lattice covering density. Forcade–Lamoreaux conjecture. … The value is supported by…

For each $n\geq 6$, let $LD_n$ be the lattice defined in Dutour (2005), and let a covering maximum be a lattice whose covering density decreases under perturbation. The covering-de…

Let $K$ be a two-dimensional convex domain. The covering densities $\theta^t(K)$ and $\theta^l(K)$ denote, respectively, the least translative covering density and the least lattic…

Dougherty–Faber conjecture. The optimal lattice covering density is