Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T06:25:55.746Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: General convexity Metric 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 Economics MTA-BME Stochastics Research Group (04118), Müegyetem rkp. 3, Budapest H-1111, Hungary email [email protected]
Gábor Somlai
Affiliation:
Department of Algebra and Number Theory, Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest H-1117, Hungary email [email protected]
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Borel, A., On free subgroups of semisimple groups. Enseign. Math. (2) 29 1983, 151–164.Google Scholar
Dekker, T. J., Decomposition of sets and spaces I, II, III. Indag. Math. 18 1956, 581–595; 19 (1957), 104–107.CrossRefGoogle Scholar
Deligne, P. and Sullivan, D., Division algebras and the Hausdorff–Banach–Tarski paradox. Enseign. Math. (2) 29(1–2) 1983, 145–150.Google Scholar
van Douwen, E., Indivisibility of balls in Euclidean n-space. Topology Appl. 51(2) 1993, 183–185.CrossRefGoogle Scholar
Kiss, G. and Laczkovich, M., Decomposition of balls into congruent pieces. Mathematika 57(1) 2011, 89–107.CrossRefGoogle Scholar
Richter, C., Affine divisibility of convex sets. Bull. Lond. Math. Soc. 41(4) 2009, 757–768.CrossRefGoogle Scholar
van der Waerden, B. L., Aufgaben 51. Elem. Math. 4 1949, 18.Google Scholar
Wagon, S., Partitioning intervals, spheres and balls into congruent pieces. Canad. Math. Bull. 26(3) 1983, 337–340.CrossRefGoogle Scholar
Wagon, S., The Banach–Tarski paradox, 2nd edn., Cambridge University Press (1986); first paperback edition, 1993.Google Scholar