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A continuous movement version of the Banach-Tarski paradox: a solution to de Groot’s problem. (English) Zbl 1134.03028

Summary: In 1924 Banach and Tarski demonstrated the existence of a paradoxical decomposition of the 3-ball \(B\), i.e., a piecewise isometry from \(B\) onto two copies of \(B\). This article answers a question of de Groot from 1958 [see S. Wagon, The Banach-Tarski paradox. Cambridge etc.: Cambridge University Press (1985; Zbl 0569.43001)] by showing that there is a paradoxical decomposition of \(B\) in which the pieces move continuously while remaining disjoint to yield two copies of \(B\). More generally, we show that if \(n \geq 2\), any two bounded sets in \(\mathbb{R}^n\) that are equidecomposable with proper isometries are continuously equidecomposable in this sense.

MSC:

03E25 Axiom of choice and related propositions
28E15 Other connections with logic and set theory

Citations:

Zbl 0569.43001
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References:

[1] The Banach-Tarski paradox 24 (1985)
[2] Rendiconti dell’lstituto di Matematica dell’Università di Trieste 23 pp 145– (1991)
[3] DOI: 10.1073/pnas.89.22.10726 · Zbl 0768.04002
[4] Classical descriptive set theory 156 (1995) · Zbl 0819.04002
[5] Journal fÜr die Reine und Angewandte Mathematik 404 pp 77– (1990)
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