Coverings by Homothetic Bodies - Illumination and Related Topics

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 ℇ \(\mathbb{E}^d\) \ 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.