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PARADOXICAL DECOMPOSITIONS OF THE CUBE AND INJECTIVITY J. D. MAITLAND WRIGHT Introduction The Banach-Tarski Paradox for the cube states that the open unit cube in three dimensions can be decomposed into finitely many pieces, which can then, by rotation and translation, be re-assembled into two copies of the unit cube. That is, there exist disjoint sets (A A ,...,A , B...,B ) whose union is (0,1) and there exist Euclidean lt 2 n lt m isometries (p p ,...,p , a ,...,a ) such that v 2 n 1 m Clearly the pieces (A^...,A , if^...,i? ) cannot all be Lebesgue measurable, n m because Lebesgue measure is preserved by rotation and translation. The problem of Marzewski for the cube (which is some sixty years old) asks: Does there exist a Banach-Tarski paradoxical decomposition of the unit cube, in which each piece has the Baire Property! Wagon [9, p. 30] points out: 'Although little is known about this question, it seems reasonable to conjecture that the answer is no. First of all, it is unlikely that the proof of the Banach-Tarski Paradox, which uses the Axiom of Choice to define the pieces, could be modified so that the pieces
Bulletin of the London Mathematical Society – Wiley
Published: Jan 1, 1990
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