by C. Richter
Preprint series: 99-37, Reports on Algebra and Geometry
The paper is published: Discrete Comput. Geom 25, 65-83, 2001.
Abstract: The paradox of Banach, Tarski and Hausdorff shows that any two
bounded sets $M,N \subseteq E^3$ with non-empty interior are
equidecomposable. The result remains true if $M$ and $N$ are
replaced by collections of sets. We present quantified versions
of the paradox by giving estimates for the minimal number of
pieces in such decompositions. The emphasis is on replications
of sets $M$, i.e., on the equidecomposability of $M$ with $k$
copies of $M$, $k \ge 2$. In particular, we discuss the problem
of replicating the cube.
Keywords: Banach-Tarski paradox, equidecomposable, degree of equidecomposability, replicating sets, duplicating the cube