Decomposition of convex figures into similar pieces
\textit{G. Valette} and \textit{T. Zamfirescu} [J. Comb. Theory, Ser. B 16, 1- 16 (1974; Zbl 0267.05029)] stated without proof that a convex figure \(P\) is a polygon if it is the union of finitely many (but at least two) nonoverlapping and congruent sets similar to \(P\). In the above statement ``convex figure'' is used to indicate a convex compact subset of the plane having a nonempty interior. In this paper two generalizations of this statement are given. The author first proves that if the convex figure \(P\) is the union of finitely many (but at least two) nonoverlapping convex figures such that one of them is similar to \(P\), then \(P\) is a polygon. An analogous result is obtained if the restriction that \(P\) be the union of finitely many nonoverlapping convex figures such that at least one of them is similar to \(P\) is replaced by the assumption that \(P\) is the union of infinitely many nonoverlapping sets all similar to \(P\). The paper contains several remarks, problems and an example that shows whereas a polygon \(P\) that can be decomposed into finitely many nonoverlapping sets similar to \(P\) can also be decomposed into infinitely many such sets, the converse need not necessarily hold.
scientific article; zbMATH DE number 735340