The Banach-Tarski theorem on polygons, and the cancellation law

Authors:
R. J. Gardner and M. Laczkovich

Journal:
Proc. Amer. Math. Soc. **109** (1990), 1097-1102

MSC:
Primary 52A10; Secondary 28C10, 52A20

MathSciNet review:
1017001

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Abstract: The Banach-Tarski theorem on polygons in implies that two polygons are equidecomposable if and only if they are equidissectable. The possibility of strengthening this result in various ways is investigated. We show that if two polytopes in are equidecomposable under a finite set of isometries which generates a discrete group, then they are equidissectable using the same isometries. We then give a simple example in showing that this is not true for arbitrary finite sets of isometries. A modification of this example is used to answer a question of S. Wagon concerning the cancellation law.

**[1]**S. Banach and A. Tarski,*Sur la décomposition des ensembles des points en parties respectivement congruentes*, Fund. Math.**6**(1924), 244-277.**[2]**R. J. Gardner,*Convex bodies equidecomposable by locally discrete groups of isometries*, Mathematika**32**(1985), no. 1, 1–9. MR**817100**, 10.1112/S0025579300010780**[3]**R. J. Gardner,*A problem of Sallee on equidecomposable convex bodies*, Proc. Amer. Math. Soc.**94**(1985), no. 2, 329–332. MR**784187**, 10.1090/S0002-9939-1985-0784187-9**[4]**M. Laczkovich,*Closed sets without measurable matching*, Proc. Amer. Math. Soc.**103**(1988), no. 3, 894–896. MR**947676**, 10.1090/S0002-9939-1988-0947676-2**[5]**M. Laczkovich,*Equidecomposability and discrepancy; a solution of Tarski’s circle-squaring problem*, J. Reine Angew. Math.**404**(1990), 77–117. MR**1037431**, 10.1515/crll.1990.404.77**[6]**-,*Invariant signed measures and the cancellation law*, preprint.**[7]**L. Lovász and M. D. Plummer,*Matching theory*, North-Holland, Amsterdam, 1986.**[8]**G. T. Sallee,*Research Problems: Are Equidecomposable Plane Convex Sets Convex Equidecomposable?*, Amer. Math. Monthly**76**(1969), no. 8, 926–927. MR**1535587**, 10.2307/2317952**[9]**J. K. Truss,*The failure of cancellation laws for equidecomposability types*, Canad. J. Math.**42**(1990), no. 4, 590–606. MR**1074225**, 10.4153/CJM-1990-031-5**[10]**Stan Wagon,*The Banach-Tarski paradox*, Encyclopedia of Mathematics and its Applications, vol. 24, Cambridge University Press, Cambridge, 1985. With a foreword by Jan Mycielski. MR**803509**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-1017001-9

Keywords:
Polygon,
convex body,
equidecomposable,
equidissectable,
circlesquaring,
matching,
cancellation law

Article copyright:
© Copyright 1990
American Mathematical Society