Each convex body in \(E^3\) symmetric about a plane can be illuminated by 8 directions
Let \(C\) be a convex body in three-dimensional Euclidean space \(E^3\). A direction \(v\) is said to illuminate a point \(x\) of the boundary of \(C\) if a ray, emanating from \(x\) and having direction \(v\), meets an interior point of \(C\). A set of directions illuminate the body \(C\) if each point of the boundary is illuminated by at least one of the given directions. Hadwiger conjectured that each convex body in \(E^d\) can be illuminated by \(2^d\) directions. This conjecture is true in the plane and was proved to be true for centrally symmetric convex bodies in \(E^3\) by \textit{M. Lassak} [J. Lond. Math. Soc. 30, No. 2, 501-511 (1984; Zbl 0585.52007)]. In this paper it is shown that each convex body in \(E^3\) symmetric about a plane can be illuminated by eight directions. The basic idea in the proof is to use four directions illuminating the intersection \(B\) of \(C\) with the plane of symmetry and build up for each of these directions two directions illuminating a neighborhood of \(B\) in the boundary of \(C\). This procedure requires numerous elaborated adjustments depending on the shape of \(B\).
scientific article; zbMATH DE number 1565891
scientific article