Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-10T05:00:29.927Z Has data issue: false hasContentIssue false
  • English
  • Français

A SPIKY BALL

Part of: Discrete geometry General convexity

Published online by Cambridge University Press:  17 February 2016

Márton Naszódi
Márton Naszódi*
Affiliation:
ELTE, Department of Geometry, Lorand Eötvös University, Pázmány Péter Sétány 1/C, Budapest 1117, Hungary email marton.naszodi@math.elte.hu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The illumination problem may be phrased as the problem of covering a convex body in Euclidean $n$ -space by a minimum number of translates of its interior. By a probabilistic argument, we show that, arbitrarily close to the Euclidean ball, there is a centrally symmetric convex body of illumination number exponentially large in the dimension.

MSC classification

Primary: 52A22: Random convex sets and integral geometry 52C17: Packing and covering in $n$ dimensions

Information

Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016 , pp. 630 - 636
Copyright
Copyright © University College London 2016 

References

Bezdek, K., The illumination conjecture and its extensions. Period. Math. Hungar. 53(1–2) 2006, 59–69.CrossRefGoogle Scholar
Bezdek, K., Classical Topics in Discrete Geometry (CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC), Springer (New York, 2010).CrossRefGoogle Scholar
Bezdek, K. and Kiss, G., On the X-ray number of almost smooth convex bodies and of convex bodies of constant width. Canad. Math. Bull. 52(3) 2009, 342–348.CrossRefGoogle Scholar
Bezdek, K. and Litvak, A. E., On the vertex index of convex bodies. Adv. Math. 215(2) 2007, 626–641.CrossRefGoogle Scholar
Boltyanski, V., The problem of illuminating the boundary of a convex body. Izv. Mold. Fil. Akad. Nauk SSSR 76 1960, 77–84.Google Scholar
Boltyanski, V., Martini, H. and Soltan, P. S., Excursions into Combinatorial Geometry (Universitext), Springer (Berlin, 1997); MR 1439963 (98b:52001).CrossRefGoogle Scholar
Böröczky, K. Jr. and Wintsche, G., Covering the sphere by equal spherical balls. In Discrete and Computational Geometry (Algorithms and Combinatorics 25 ), Springer (Berlin, 2003), 235–251.CrossRefGoogle Scholar
Brass, P., Moser, W. and Pach, J., Research Problems in Discrete Geometry, Springer (New York, 2005).Google Scholar
Gluskin, E. D. and Litvak, A. E., A remark on vertex index of the convex bodies. In Geometric Aspects of Functional Analysis (Lecture Notes in Mathematics 2050 ), Springer (Heidelberg, 2012), 255–265.CrossRefGoogle Scholar
Hadwiger, H., Ungelöste probleme, nr. 38. Elem. Math. 15 1960, 130–131.Google Scholar
Martini, H. and Soltan, V., Combinatorial problems on the illumination of convex bodies. Aequationes Math. 57(2–3) 1999, 121–152.CrossRefGoogle Scholar
Mitzenmacher, M. and Upfal, E., Probability and Computing: Randomized Algorithms and Probabilistic Analysis, Cambridge University Press (Cambridge, 2005).CrossRefGoogle Scholar
Rogers, C. A., Covering a sphere with spheres. Mathematika 10 1963, 157–164.CrossRefGoogle Scholar