Fractional illumination of convex bodies

Abstract

We introduce a fractional version of the illumination problem of Gohberg, Markus, Boltyanski and Hadwiger, according to which every convex body in ${\mathbb R}^d$ is illuminated by at most $2^d$ directions. We say that a weighted set of points on ${\mathbb S}^{d-1}$ 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 $2^d$, and of a general convex body $\binom{2d}{d}$. As a corollary, we obtain that for any o-symmetric convex polytope with $k$ vertices, there is a direction that illuminates at least $\left\lceil\frac{k}{2^d}\right\rceil$ vertices.