Multiple Lattice Packings and Coverings of the Plane with Triangles

Given a convex disk $K$ and a positive integer $j$, let $δ_L^j(K)$ and $\vartheta_L^j(K)$ denote the $j$-fold lattice packing density and the $j$-fold lattice covering density of $K$, respectively. I will prove that for every triangle $T$ we have that $δ_L^j(T)=\frac{2j^2}{2j+1}$ and $\vartheta_L^j(