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DECOMPOSITION OF BALLS INTO CONGRUENT PIECES

Part of: General convexity Metric geometry

Published online by Cambridge University Press:  21 January 2011

Gergely Kiss  and
Miklós Laczkovich
Gergely Kiss
Affiliation:
Department of Analysis, Eötvös Loránd University, Budapest, Pázmány Péter sétány 1/C 1117, Hungary (email: kisss@cs.elte.hu)
Miklós Laczkovich
Affiliation:
Department of Analysis, Eötvös Loránd University, Budapest, Pázmány Péter sétány 1/C 1117, Hungary Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, U.K. (email: laczk@cs.elte.hu)
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Abstract

We prove that if 3|d, then the d-dimensional balls are m-divisible for every m large enough. In particular, the three-dimensional balls are m-divisible for every m≥22.

MSC classification

Secondary: 52A15: Convex sets in $3$ dimensions (including convex surfaces) 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 51F20: Congruence and orthogonality

Information

Type
Research Article
Information
Mathematika , Volume 57 , Issue 1 , January 2011 , pp. 89 - 107
Copyright
Copyright © University College London 2011

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References

[1]Deligne, P. and Sullivan, D., Division algebras and the Hausdorff–Banach–Tarski paradox. Enseign. Math. (2) 29(1–2) (1983), 145–150.Google Scholar
[2]de Groot, J., Orthogonal isomorphic representations of free groups. Canad. J. Math. 8 (1956), 256–262.CrossRefGoogle Scholar
[3]Edelstein, M., Isometric decompositions. J. London Math. Soc. (2) 37 (1988), 158–163.CrossRefGoogle Scholar
[4]Laczkovich, M., Paradoxical decompositions using Lipschitz functions. Mathematika 39 (1992), 216–222.Google Scholar
[5]Prasolov, V. V., Problems and Theorems in Linear Algebra (Translations of Mathematical Monographs 134), American Mathematical Society (Providence, RI, 1994).CrossRefGoogle Scholar
[6]Richter, C., Most convex bodies are isometrically indivisible. J. Geom. 89 (2008), 130–137.CrossRefGoogle Scholar
[7]Richter, C., Affine divisibility of convex sets. Bull. London Math. Soc. 41(4) (2009), 757–768.CrossRefGoogle Scholar
[8]Wagon, S., Partitioning intervals, spheres and balls into congruent pieces. Canad. Math. Bull. 26(3) (1983), 337–340.CrossRefGoogle Scholar
[9]Wagon, S., The Banach–Tarski Paradox, 2nd edn., Cambridge University Press (Cambridge, 1986), First paperback edition, 1993.Google Scholar