8 problems

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.…

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…

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…

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

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…