Abstract
If a convex plane figureP can be decomposed into finitely many nonoverlapping convex figures such that one of these pieces is similar toP, thenP is a polygon. Also, ifP can be decomposed into infinitely many nonoverlapping sets such that each of the pieces is similar toP, thenP is a polygon.
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References
G. Valette and T. Zamfirescu, Les partages d'un polygone convexe en 4 polygones semblambes au premier.J. Combin. Theory Ser. B. 16 (1974), 1–16.
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Communicated by Imre Bárány
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Laczkovich, M. Decomposition of convex figures into similar pieces. Discrete Comput Geom 13, 143–148 (1995). https://doi.org/10.1007/BF02574033
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DOI: https://doi.org/10.1007/BF02574033