A local characterization of combinatorial multihedrality in tilings

A locally finite face-to-face tiling of euclidean $d$-space by convex polytopes is called {\em combinatorially multihedral\/} if its combinatorial automorphism group has only finitely many orbits on the tiles. The paper describes a local characterization of combinatorially multihedral tilings in terms of centered coronas. This generalizes the Local Theorem for Monotypic Tilings, established in \cite{dolsch}, which characterizes the case of combinatorial tile-transitivity