The local minima in the lattice-simplex covering problem (Q2927764)

The paper is devoted to coverings of the space \(\mathbb{R}^{n}\) with lattice translates of an \(n\)-simplex. The problem under investigation is to determine minimal density of such coverings.NEWLINENEWLINEFor the space \(\mathbb{R}^2\) the problem is completely solved. In the space \(\mathbb{Z}^3\) a local minimum corresponding to the 84-shape was found independently by several authors using computer search (for the simplex which is the convex hull of the unit basis vectors).NEWLINENEWLINEIn the present paper, the authors define the concept of combinatorial type and propose a search algorithm which determines the local minima within an arbitrary combinatorial type.