The illumination conjecture for spindle convex bodies

A subset of the $d$-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent $d$-dimensional closed balls. A spindle convex body is called a “fat” one if it contains the centers of its generating balls. The main result of this paper is a proof of the illumination conjecture for “fat” spindle convex bodies in dimensions greater than or equal to 15.