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Decomposition of sets of small or large boundary

Part of: Combinatorics

Published online by Cambridge University Press:  26 February 2010

Miklós Laczkovich
Miklós Laczkovich
Affiliation:
Department of Analysis, Eōtvōs Loránd University, Múzeum Krt, 6-8, Budapest, Hungary
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Extract

We shall say that the sets A, B ⊂ Rk are equivalent, if they are equidecomposable using translations; that is, if there are finite decompositions and vectors x1,…, xd∈Rk such that Bj = Aj + xj, (j = 1,…,d). We shall denote this fact by In [3], Theorem 3 we proved that if A ⊂ Rk 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

Secondary: 05A18: Partitions of sets
Type
Research Article
Information
Mathematika , Volume 40 , Issue 2 , December 1993 , pp. 290 - 304
Copyright
Copyright © University College London 1993

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References

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