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Convex bodies equidecomposable by locally discrete groups of isometries

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

R. J. Gardner
R. J. Gardner
Affiliation:
University of Petroleum and Minerals, Department of Mathematical Sciences, Dhahran, Saudi Arabia.
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Abstract

We show that if a polytope K1, in ℝd can be partitioned into a finite number of sets, and these sets can be moved by isometries in a locally discrete group to form a convex body K2, then K2 is a polytope and a similar partition can be made where the sets involved are simplices with disjoint interiors. This gives partial answers to questions of Tarski, Sallee and Wagon.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 32 , Issue 1 , June 1985 , pp. 1 - 9
Copyright
Copyright © University College London 1985

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References

1.Banach, S. and Tarski, A.. Sur la décomposition des ensembles des points en parties respectivement congruentes. Fund. Math., 6 (1924), 244–277.CrossRefGoogle Scholar
2.Benson, C. T. and Grove, L. C.. Finite Reflection Groups (Bogden and Quigley, New York, 1971).Google Scholar
3.Boltianskii, V.. Hubert's Third Problem (Winston, Washington, 1978).Google Scholar
4.Dubins, L., Hirsch, M. and Karush, J.. Scissor congruence. Israel J. Math., 1 (1963), 239–247.CrossRefGoogle Scholar
5.Gardner, R. J.. A problem of Sallee on equidecomposable convex bodies. Proc. Amer. Math. Soc. To appear.Google Scholar
6.Mycielski, J.. Finitely additive invariant measures, I. Colloq. Math, 42 (1979), 309–318.CrossRefGoogle Scholar
7.Mycielski, J. and Wagon, S.. Large free groups of isometries. L'Enseignement Math., 30 (1984), 247–267.Google Scholar
8.Sallee, G. T.. Are equidecomposable plane convex sets convex equidecomposable? Amer. Math. Monthly, 76 (1969), 926–927.CrossRefGoogle Scholar
9.Tarski, A.. Probleme 38. Fund. Math., 7 (1925), 381.Google Scholar
10.Wagon, S.. The Banach-Tarski Paradox (Cambridge University Press, New York, 1985).CrossRefGoogle Scholar