Fractional illumination of convex bodies. (English) Zbl 1189.52005
Summary: 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 \(\lceil\frac{k}{2^d}\rceil\) vertices.
MSC:
52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |
52A35 | Helly-type theorems and geometric transversal theory |
52C45 | Combinatorial complexity of geometric structures |