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

A family of n -dimensional Lee spheres L is a tiling of R n , if ∪ L = R n and for every L u , L v ∈ L , the intersection L u ∩ L v is contained in the boundary of L u . If neighboring Lee spheres meet along entire ( n − 1 ) -dimensional faces, then L is called a face-to-face tiling. We prove the nonexistence of a face-to-face tiling of R 4 with Lee spheres of different radii.