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Decomposition of sets of small or large boundary

Published online by Cambridge University Press:  26 February 2010

Miklós Laczkovich
Affiliation:
Department of Analysis, Eōtvōs Loránd University, Múzeum Krt, 6-8, Budapest, Hungary
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We shall say that the sets A, BRk are equivalent, if they are equidecomposable using translations; that is, if there are finite decompositions and vectors x1,…, xdRk such that Bj = Aj + xj, (j = 1,…,d). We shall denote this fact by In [3], Theorem 3 we proved that if ARk is a bounded measurable set of positive measure then A is equivalent to a cube provided that Δ(δA)<k where δA denotes the boundary of A and Δ(E) denotes the packing dimension (or box dimension or upper entropy index) of the bounded set E. This implies, in particular, that any bounded convex set of positive measure is equivalent to a cube. C. A. Rogers asked whether or not the set

MSC classification

Type
Research Article
Copyright
Copyright © University College London 1993

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