The illumination conjecture and its extensions

The Illumination Conjecture was raised independently by Boltyanski and Hadwiger in 1960. According to this conjecture any d-dimensional convex body can be illuminated by at most 2d light sources. This is an important fundamental problem. The paper surveys the state of the art of the Illumination Conjecture. 1 1 The Illumination Conjecture Let K be a convex body (i.e. a compact convex set with nonempty interior) in the ddimensional Euclidean space E, d ≥ 2. According to Hadwiger [22] an exterior point p ∈ E d \K of K illuminates the boundary point q of K if the half line 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 [22] raised the following amazingly elementary but, very fundamental question. An equivalent but somewhat different looking concept of illumination was introduced by Boltyanski in [13]. There he proposed to use directions (i.e. unit vectors) instead of point sources for the illumination of convex bodies. Based on these circumstances the following conjecture we call the Boltyanski-Hadwiger Illumination Conjecture. ∗The author was partially supported by the Hung. Nat. Sci. Found (OTKA), grant no. T043556 and by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. 2000 Mathematics Subject Classification: 52C17, 52B11; Key words, phrases: illumination by affine subspaces, covering by homothets.