Naszódi, Márton and Pach, János and Swanepoel, K. (2017) ARRANGEMENTS OF HOMOTHETS OF A CONVEX BODY. MATHEMATIKA, 63 (2). pp. 696-710. ISSN 0025-5793
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Abstract
Answering a question of Furedi and Loeb [On the best constant for the Besicovitch covering theorem, Proc. Amer: Math, Soc. 121(4) (1994), 10631073], we show that the maximum number of pairwise intersecting homothets of a d-dimensional centrally symmetric convex body K. none of which contains the center of another in its interior, is at most 0(3(d) log d). If K is not necessarily centrally symmetric and the role of its center is played by its centroid, then the above bound can be replaced by 0 (3(d) (2dd)d log d). We establish analogous results for the case where the center is defined as an arbitrary point in the interior of K. We also show that, in the latter case, one can always find families of at least Omega ((2/root 3)(d)) translates of K with the above property.
| Item Type: | Article |
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| Additional Information: | OA green_published Department of Geometry, Lorand Eötvös University, Pazmány Péter Sétany 1/C, Budapest, 1117, Hungary Department of Mathematics, London School of Economics and Political Science, Houghton Street, London, WC2A 2AE, United Kingdom École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland Rényi Institute, Budapest, Hungary Cited By :2 Export Date: 10 January 2019 Funding details: École Polytechnique Fédérale de Lausanne, EPFL Funding details: Magyar Tudományos Akadémia, MTA Funding details: Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung, SNF, 200020-144531 Funding details: Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung, SNF, 200020-162884 Funding details: Nemzeti Kutatási, Fejlesztési és Innovációs Hivatal Funding details: PD104744 Funding details: K119670 Funding text 1: MSC (2010): 52C17 (primary), 46B06, 52A23 (secondary). Márton Naszódi acknowledges the support of the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and of the National Research, Development and Innovation Office, NKFIH grants PD104744 and K119670. Part of this paper was written when Swanepoel visited EPFL in April 2015. Research by János Pach was supported in part by the Swiss National Science Foundation, grants 200020-144531 and 200020-162884. Funding Agency and Grant Number: Janos Bolyai Research Scholarship of the Hungarian Academy of Sciences; National Research, Development and Innovation Office, NKFIH [PD104744, K119670]; Swiss National Science Foundation [200020-144531, 200020-162884] Funding text: Marton Naszodi acknowledges the support of the Janos Bolyai Research Scholarship of the Hungarian Academy of Sciences and of the National Research, Development and Innovation Office, NKFIH grants PD104744 and K119670. Part of this paper was written when Swanepoel visited EPFL in April 2015. Research by Janos Pach was supported in part by the Swiss National Science Foundation, grants 200020-144531 and 200020-162884. |
| Uncontrolled Keywords: | SPACES; BODIES; SPHERE; |
| Subjects: | Q Science / természettudomány > QA Mathematics / matematika |
| SWORD Depositor: | MTMT SWORD |
| Depositing User: | MTMT SWORD |
| Date Deposited: | 22 Sep 2019 14:31 |
| Last Modified: | 22 Sep 2019 14:31 |
| URI: | http://real.mtak.hu/id/eprint/100308 |
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