Illuminating spiky balls and cap bodies

TL;DR

This paper establishes upper bounds on the illumination numbers of certain convex bodies called spiky balls and cap bodies, proving the Illumination Conjecture for high-dimensional centrally symmetric cases.

Contribution

It proves the Illumination Conjecture for high-dimensional centrally symmetric cap bodies and improves bounds for 1-unconditionally symmetric cases.

Findings

Centrally symmetric cap bodies in dimensions ≥20 can be illuminated by fewer than 2^d directions.

The paper confirms the Illumination Conjecture for these bodies in high dimensions.

Strengthens results for 1-unconditionally symmetric 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 $d\geq 20$. Furthermore, we strengthen the latter result for $1$-unconditionally symmetric cap bodies.