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

Abstract

A family of $n$-dimensional Lee spheres $\mathcal{L}$ is a tiling of ${\mathbb{R}}^n$, if $\cup \mathcal{L}={\mathbb{R}}^n$ and for every $L_u,L_v\in \mathcal{L}$, the intersection $L_u\cap L_v$ is contained in the boundary of $L_u$. If neighboring Lee spheres meet along entire $(n-1)$-dimensional faces, then $\mathcal{L}$ is called a face-to-face tiling. We prove the nonexistence of a face-to-face tiling of ${\mathbb{R}}^4$ with Lee spheres of different radii.