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 Rd is illuminated by at most 2d 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 prove that the fractional illumination number of any o-symmetric convex body is at most 2d, and of a general convex body (2dd). As a corollary, we obtain that for any o-symmetric convex polytope with k vertices, there is a direction that illuminates at least ⌈k2d⌉ vertices.