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Coverings by Homothetic Bodies - Illumination and Related Topics

  • Károly Bezdek
Chapter
Part of the CMS Books in Mathematics book series (CMSBM)

Abstract

Let K be a convex body (i.e., a compact convex set with nonempty interior) in the d-dimensional Euclidean space \(\mathbb{E}^d\), d ≥ 2. According to Hadwiger [155] an exterior point p\(\mathbb{E}^d\) \ K of K illuminates the boundary point q of K if the haline emanating from p passing through q intersects the interior of K (at a point not between p and q). Furthermore, a family of exterior points of K say, p1; p2;…; pn illuminates K if each boundary point of K is illuminated by at least one of the point sources p1; p2;…; pn. Finally, the smallest n for which there exist n exterior points of K that illuminate K is called the illumination number of K denoted by I(K). In 1960, Hadwiger [155] raised the following amazingly elementary, but very fundamental question. An equivalent but somewhat different-looking concept of illumination was introduced by Boltyanski in [78]. There he proposed to use directions (i.e., unit vectors) instead of point sources for the illumination of convex bodies. Based on these circumstances we call the following conjecture the Boltyanski-Hadwiger Illumination Conjecture.

Keywords

Convex Body Constant Width Gauss Image Symmetric Convex Body Exterior Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

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