Abstract
The convex hull of a ball with an exterior point is called a spike (or cap). A union of finitely many spikes of a ball is called a spiky ball. If a spiky ball is convex, then we call it a cap body. In this note we upper bound the illumination numbers of 2-illuminable spiky balls as well as centrally symmetric cap bodies. In particular, we prove the Illumination Conjecture for centrally symmetric cap bodies in sufficiently large dimensions. In fact, we do a bit more by showing that any d-dimensional centrally symmetric cap body can be illuminated by <2d directions in Euclidean d-space for d=3,4,9 and d≥19. Furthermore, we strengthen the latter result for 1-unconditionally symmetric cap bodies.
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