  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 $I(K_c)\leq6$ for
centrally symmetric cap bodies of a ball in $\mathbb{E}^3$, and $I(K_c)\leq 8$
for unconditionally symmetric cap bodies of a ball in $\mathbb{E}^4$.
