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on Hadwiger covering / illumination number in $\mathbb{R}^3$ ($C_{39}$)

The statement

$C_{39}=H_3$ is the Hadwiger covering number in dimension $3$ , which can also be formulated in terms of illumination of the boundary. [ABP2024-equivalence-illumination]

Given sets $K,L\subset \mathbb{R}^n$ , let $C(K,L)$ be the minimal number of translates of $L$ needed to cover $K$ . [ABP2024-def-CKL]

For a convex body $K\subset \mathbb{R}^n$ , write $\operatorname{int}(K)$ for its interior. The Hadwiger covering number in dimension $n$ is the minimal number $H_n$ such that any $n$ -dimensional convex body can be covered by $H_n$ translates of its interior. [ABP2024-def-Hn]

The constant of interest here is $H_3$ . [ABP2024-def-Hn]

For symmetric convex bodies one also considers the symmetric covering number $H_n^s$ , defined analogously. [ABP2024-def-Hns]

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.