Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T13:26:59.912Z Has data issue: false hasContentIssue false
  • English
  • Français

On the X-ray Number of Almost Smooth Convex Bodies and of Convex Bodies of Constant Width

Published online by Cambridge University Press:  20 November 2018

K. Bezdek  and
Gy. Kiss
K. Bezdek
Affiliation:
Department of Mathematics and Statistics, 2500 University Drive N.W., University of Calgary, AB, T2N 1N4 e-mail: [email protected]
Gy. Kiss
Affiliation:
Department of Geometry, Mathematical Institute, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary e-mail: [email protected]
Rights & Permissions [Opens in a new window]

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 X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions 3, 4, 5 and 6.

Keywords

52A2052A3752C1752C35almost smooth convex bodyconvex body of constant widthweakly neighbourly antipodal convex polytopeIllumination ConjectureX-ray numberX-ray Conjecture
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 3 , 01 September 2009 , pp. 342 - 348
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Bezdek, K., The problem of illumination of the boundary of a convex body by affine subspaces. Mathematika 38(1991), 362–375.Google Scholar
[2] Bezdek, K., The illumination conjecture and its extensions. Period. Math. Hungar. 53(2006), 59–69.Google Scholar
[3] Bezdek, K. and Zamfirescu, T., A characterization of 3-dimensional convex sets with an infinite X-ray number. In: Intuitive Geometry, Colloq. Math. Soc. J. Bolyai 63. North-Holland, Amsterdam 1994, pp. 33–38.Google Scholar
[4] Bezdek, K., Naszódi, M., Lángi, Zs., and Papez, P., Ball-Polyhedra. Discrete Comput. Geom. 38(2007), 201–230.Google Scholar
[5] Böröczky, K. Jr., Finite Packing and Covering., Cambridge Tracts in Mathematics 154. Cambridge University Press, Cambridge, 2004.Google Scholar
[6] Danzer, L. and Grünbaum, B., Über zwei Probleme bezüglich konvexer Körper von P. Erdös und von V. L. Klee. Math. Z. 79(1962), 95–99.Google Scholar
[7] Dekster, B. V., The Jung theorem for spherical and hyperbolic spaces. Acta Math. Hungar. 67(1995), 315–331.Google Scholar
[8] Tóth, G. Fejes, Kreisüberdeckungen der Sphäre. Studia Sci.Math. Hungar. 4(1969), 225–247.Google Scholar
[9] Kiss, Gy., Illumination problems and codes. Periodica Math. Hungar. 39(1999), 65–71.Google Scholar
[10] Martini, H. and Soltan, V., Combinatorial problems on the illumination of convex bodies. Aequationes Math. 57(1999), 121–152.Google Scholar
[11] Schramm, O., Illuminating sets of constant width. Mathematika 35(1988), 180–189.Google Scholar
[12] Schürmann, A. and Swanepoel, K. J., Three-dimensional antipodal and norm-equilateral sets. Pacific J. Math. 228(2006), 349–370.Google Scholar
[13] Swanepoel, K. J., Quantitative illumination of convex bodies and vertex degrees of geometric Steiner minimal trees. Mathematika 52(2005), 47–52.Google Scholar