On the illumination of centrally symmetric cap bodies in small dimensions

The illumination number $I(K)$ of a convex body $K$ in Euclidean space ${\mathbb{E}}^d$ is the smallest number of directions that completely illuminate the boundary of a convex body. A cap body $K_c$ of a ball is the convex hull of a Euclidean ball and a countable set of points outside the ball under the condition that each segment connecting two of these points intersects the ball. The main results of this paper are the sharp estimates $I(K_c)\le 6$ for centrally symmetric cap bodies of a ball in ${\mathbb{E}}^3$, and $I(K_c)\le 8$ for unconditionally symmetric cap bodies of a ball in ${\mathbb{E}}^4$.