Illuminating spiky balls and cap bodies

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 by showing that any d-dimensional centrally symmetric cap body can be illuminated by < 2 d directions in Euclidean d-space for all d ≥ 20. Furthermore, we strengthen the latter result for 1-unconditionally symmetric cap bodies.