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DECOMPOSITION OF BALLS IN $\mathbb{R}^{d}$

Part of: Metric geometry General convexity

Published online by Cambridge University Press:  22 January 2016

Gergely Kiss  and
Gábor Somlai
Gergely Kiss
Affiliation:
Department of Stochastics, Faculty of Natural Sciences, Budapest University of Technology and Economics MTA-BME Stochastics Research Group (04118), Müegyetem rkp. 3, Budapest H-1111, Hungary email kisss@cs.elte.hu
Gábor Somlai
Affiliation:
Department of Algebra and Number Theory, Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest H-1117, Hungary email gsomlai@cs.elte.hu
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Abstract

We investigate the problem of the decomposition of balls into a finite number of congruent pieces in dimension $d=2k$. In addition, we prove that the $d$-dimensional unit ball $B_{d}$ can be divided into a finite number of congruent pieces if $d=4$ or $d\geqslant 6$. We show that the minimal number of required pieces is less than $20d$, if $d\geqslant 10$.

MSC classification

Primary: 52A15: Convex sets in $3$ dimensions (including convex surfaces) 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 51F20: Congruence and orthogonality
Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016 , pp. 378 - 405
Copyright
Copyright © University College London 2016 

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