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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 spikyball is convex, then we call it a cap body. In this note we upper bound theillumination numbers of $2$-illuminable spiky balls as well as centrallysymmetric cap bodies. In particular, we prove the Illumination Conjecture forcentrally symmetric cap bodies in sufficiently large dimensions by showing thatany $d$-dimensional centrally symmetric cap body can be illuminated by $<2^d$directions in Euclidean $d$-space for all $d\geq 20$. Furthermore, westrengthen the latter result for $1$-unconditionally symmetric cap bodies.

Illuminating spiky balls and cap bodies