Skip to main content Accessibility help

Convex bodies equidecomposable by locally discrete groups of isometries

  • R. J. Gardner (a1)


We show that if a polytope K1, in d 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.



Hide All
1.Banach, S. and Tarski, A.. Sur la décomposition des ensembles des points en parties respectivement congruentes. Fund. Math., 6 (1924), 244277.
2.Benson, C. T. and Grove, L. C.. Finite Reflection Groups (Bogden and Quigley, New York, 1971).
3.Boltianskii, V.. Hubert's Third Problem (Winston, Washington, 1978).
4.Dubins, L., Hirsch, M. and Karush, J.. Scissor congruence. Israel J. Math., 1 (1963), 239247.
5.Gardner, R. J.. A problem of Sallee on equidecomposable convex bodies. Proc. Amer. Math. Soc. To appear.
6.Mycielski, J.. Finitely additive invariant measures, I. Colloq. Math, 42 (1979), 309318.
7.Mycielski, J. and Wagon, S.. Large free groups of isometries. L'Enseignement Math., 30 (1984), 247267.
8.Sallee, G. T.. Are equidecomposable plane convex sets convex equidecomposable? Amer. Math. Monthly, 76 (1969), 926927.
9.Tarski, A.. Probleme 38. Fund. Math., 7 (1925), 381.
10.Wagon, S.. The Banach-Tarski Paradox (Cambridge University Press, New York, 1985).
MathJax is a JavaScript display engine for mathematics. For more information see

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed