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

Part of:General convexityMetric geometry

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 EconomicsMTA-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

Abstract

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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

Information

Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016, pp. 378 - 405
Copyright
Copyright © University College London 2016 

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