Illumination number of 3-dimensional cap bodies

TL;DR

The paper resolves the illumination problem for the broad class of 3-dimensional cap bodies by proving $I(K)=6$ for all such bodies, with illumination directions given by the vertices of a regular tetrahedron plus two body-dependent directions. It combines probabilistic arguments on random tetrahedral rotations with an exact integer linear programming bound to limit the number of unilluminated caps. The result tightens the 3D Hadwiger illumination conjecture for cap bodies, extends prior symmetric-case bounds, and provides a constructive illumination scheme applicable to all 3D cap bodies. The authors also include a computationalCertifiable ILP approach to certify the bound, highlighting a practical route to similar bounds in other geometric classes.

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.

Figures (2)

• Figure 1: A cap body in $\mathbb{R}^2$ with 2 vertices $x_1,x_2$.
• Figure 2: A base cap $S_i$ with its center $\widehat{x_i}$.

Theorems & Definitions (9)

• Conjecture 1
• Theorem 2
• Proposition 3
• Lemma 4
• proof
• Lemma 5
• proof
• Remark 6
• Remark 7