Illumination number of 3-dimensional cap bodies

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.

💡 Research Summary

The paper addresses the illumination number of three‑dimensional cap bodies, a special class of convex bodies formed by taking the convex hull of the unit ball B³ together with a finite set of exterior points (the “vertices”). A cap body K = conv(B³ ∪ V) is required to satisfy the condition that the line segment joining any two distinct vertices intersects B³; equivalently, the “spikes” attached to the ball have pairwise disjoint interiors. The illumination number I(K) is the smallest number of external light directions (or parallel beams) that together illuminate every boundary point of K, i.e., for each boundary point there exists a direction whose ray enters the interior of K.

Previously, the illumination conjecture (also known as Hadwiger‑Boltyanski’s covering conjecture) predicts that any convex body in ℝⁿ can be illuminated by at most 2ⁿ directions, with equality only for affine copies of the n‑cube. In dimension three the conjecture remains open in general; only special families have been settled. Notably, centrally symmetric cap bodies were shown to have I(K)=6 (Ivánov–Strachan, 2021), and the bound is sharp because a suitably scaled regular octahedron together with the unit ball yields a symmetric cap body with illumination number exactly six.

The present work extends this result to all three‑dimensional cap bodies, removing the symmetry assumption. The main theorem states:

Theorem 2. For any cap body K ⊂ ℝ³, I(K) ≤ 6. Moreover, the six illuminating directions can be chosen as the four vertices of a regular tetrahedron (inscribed in the unit sphere) together with two additional directions that depend on the specific geometry of K.

The proof proceeds in several stages:

1. Geometric Preliminaries. For each vertex x_i of K, the associated “base cap” S_i on the unit sphere is defined as the spherical cap centered at b_{x_i}=x_i/‖x_i‖ with angular radius φ_i = arccos(1/‖x_i‖). Because K is convex, the caps S_i are pairwise disjoint and each satisfies φ_i < π/2. Illuminating a cap S_i is equivalent to having a direction v that lies in the complementary cap C(−b_{x_i}, π/2−φ_i).

2. Reduction to a Finite Set of Directions. Proposition 3 (cited from earlier work) translates illumination into two conditions: (i) every complementary cap must contain at least one chosen direction, and (ii) the positive hull of the chosen directions must be the whole space ℝ³. The latter is automatically satisfied by the four vertices of any regular tetrahedron, because they positively span ℝ³.

3. Probabilistic Rotation of a Tetrahedron. Let L be the set of four tetrahedral vertices. For a random rotation R ∈ SO(3), consider the rotated set L′ = R(L). For a given angular threshold θ, define C_θ = ⋃_{l∈L} C