Nonexistence of face-to-face four-dimensional tilings in the Lee metric (Q854814)

For \(x,y\in\mathbb R^{n}\), let \(d(x,y)=| x_{1}-y_{1}| +\cdots +| x_{n}-y_{n}| \) for \(x\in\mathbb R^{n}\), let \(C(x)\) be the axis-parallel cube with center \(x\) and edge length \(1\). A Lee sphere with center \(x\in\mathbb R^{n}\) and radius \(r\) then is the set \[ L_{r}(x)=\bigcup \{ C(y):y\in\mathbb Z^{n}, d(x,y)\leq r\}. \] The main result of the article says that there does not exist a facet-to-facet tiling of \(\mathbb R^{4}\) by Lee spheres of different radii, all at least \(2\). The proof is by means of an algorithm.