Facet-to-facet implies face-to-face (Q1114937)

A locally finite convex tiling of \({\mathbb{E}}^ d\) is facet-to-facet (face- to-face) if for any two polyhedra of the tiling which have \((d-1)\)- dimensional intersection (non-empty intersection) the intersection is a facet, i.e. a face of dimension \(d-1\) (face) of both of them. It is shown that any locally finite convex tiling of \({\mathbb{E}}^ d\) which is facet to facet is also face to face and thus gives rise to a cell complex. In particular this proves that the well-known Dirichlet-Voronoi tilings and the Delone-triangulations give rise to cell complexes. [See the first author and \textit{C. G. Lekkerkerker}, Geometry of numbers (1987; Zbl 0611.10017)].