Characterization of polytopes via tilings with similar pieces

Abstract

Generalizing results by Valette, Zamfirescu and Laczkovich, we will prove that a convex body $K$ is a polytope if there are sufficiently many tilings which contain a tile similar to $K$. Furthermore, we give an example that this can not be improved.

arXiv commentaccepted for publication in Discrete and Computational Geometry