Illumination number of 3-dimensional cap bodies

TL;DR

This paper proves that all 3-dimensional cap bodies can be illuminated with six sources, confirming the illumination conjecture for this class using probabilistic and optimization methods.

Contribution

It extends the illumination number result to all 3D cap bodies, not just symmetric ones, using novel probabilistic and linear programming techniques.

Findings

All 3D cap bodies have an illumination number of 6.

Illumination directions can be chosen as vertices of a regular tetrahedron plus two special directions.

The proof employs probabilistic arguments and integer linear programming.

Abstract

The illumination conjecture asserts that any convex body in $n$-dimensional Euclidean space can be illuminated by at most $2^n$ external light sources or parallel beams of light. Despite recent progress on the illumination conjecture, it remains open in general, as well as for specific classes of bodies. Bezdek, Ivanov, and Strachan showed that the conjecture holds for symmetric cap bodies in sufficiently high dimensions. Further, Ivanov and Strachan calculated the illumination number for the class of 3-dimensional centrally symmetric cap bodies to be 6. In this paper, we show that even the broader class of all 3-dimensional cap bodies has the same illumination number 6, in particular, the illumination conjecture holds for this class. The illuminating directions can be taken to be vertices of a regular tetrahedron, together with two special directions depending on the body. The proof is based on probabilistic arguments and integer linear programming.