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 cannot be improved.
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Laczkovich, M.: Decomposition of convex figures into similar pieces. Discrete Comput. Geom. 13, 143–148 (1995)
Valette, G., Zamfirescu, T.: Les partages d’un polygone convexe en 4 polygones semblambes au premier. J. Comb. Theory, Ser. B 16, 1–16 (1974)
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Communicated by Imre Bárány.
The final preparation of this paper was supported by the Deutsche Forschungsgemeinschaft within the research training group ‘Methods for Discrete Structures’ (GRK1408).
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Adiprasito, K. Characterization of Polytopes via Tilings with Similar Pieces. Discrete Comput Geom 47, 424–429 (2012). https://doi.org/10.1007/s00454-011-9331-2
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DOI: https://doi.org/10.1007/s00454-011-9331-2