Fractional illumination of convex bodies

We introduce a fractional version of the illumination problem of Gohberg, Markus, Boltyanski and Hadwiger, according to which every convex body in R is illuminated by at most 2 directions. We say that a weighted set of points on Sd−1 illuminates a convex body K if for each boundary point of K, the total weight of those directions that illuminate K at that point is at least one. We define the fractional illumination number of K as the minimum total weight of a weighted set of points on Sd−1 that illuminates K. We prove that the fractional illumination number of any o-symmetric convex body is at most 2, and of a general convex body ` 2d d ́ . As a corollary, we obtain that for any o-symmetric convex polytope with k vertices, there is a direction that illuminates at least ̊ k 2d ˇ vertices. 1. Definitions and Results We work in the d–dimensional Euclidean space Rd, denote the origin by o and the unit sphere by Sd−1. The cardinality, interior, boundary and the volume of a set X ⊂ Rd are denoted by cardX, intX,bdX and volX, respectively. We say that a direction u ∈ Sd−1 illuminates a boundary point x of the convex body K if the ray emanating from x in the direction u intersects the interior of K. A set of directions A ⊆ Sd−1 illuminates K if each boundary point of K is illuminated by at least one member of A. The illumination number i(K) of K is the minimum number of directions that illuminate K. The following was conjectured by I. Gohberg, A. S. Markus, V. G. Boltyanski and H. Hadwiger: Every convex body in Rd is illuminated by at most 2d directions (that is, i(K) ≤ 2d) moreover, parallelotopes are the only bodies requiring 2d directions. For a thorough treatment of the development of this and related problems, see [4, 11, 14]. In this note, we introduce the following fractional version of the illumination number. Received by the editors January 14, 2009, and in revised form June 9, 2009. 2000 Mathematics Subject Classification. 52C17, 52A35, 52C45.