Hadwiger covering / illumination number in $\mathbb{R}^3$

Description of constant

$C_{39}=H_3$ 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]

Known upper bounds

Known lower bounds

Additional comments and links

• Conjectured exact value (open in dimension $3$). Hadwiger’s covering (illumination) conjecture asserts $H_n=H_n^s=2^n$ for all $n$, hence would imply $H_3=8$. [ABP2024-conj-Hn]

• Origin of the conjecture. Hadwiger posed the covering problem in 1957. [ABP2024-hadwiger-question] [Had1957]

• Centrally symmetric case in dimension $3$. The symmetric variant is known exactly: $H_3^s=8$ (and is sharp). [ABP2024-H3s-8]

• Surveys/background for the general illumination/covering problem include [ABP2024].

References

• [ABP2024] Arman, Andrii; Bondarenko, Andriy; Prymak, Andriy. On Hadwiger’s covering problem in small dimensions. Canadian Mathematical Bulletin 68(4) (2025), 1239–1250. DOI: 10.4153/S0008439525000384. Google Scholar. arXiv PDF.
  • [ABP2024-equivalence-illumination]
    loc: arXiv PDF p.1, Abstract.
    quote: “It is possible to define $H_n$ and $H_n^s$ in terms of illumination of the boundary of the body using external light sources,”
  • [ABP2024-def-CKL]
    loc: arXiv PDF p.1, Introduction (definitions paragraph).
    quote: “we denote by $C(A,B):=\min\bigl(N:\exists t_1,\dots,t_N\in\mathbb{E}^n\text{ satisfying }A\subset\bigcup_{j=1}^N(t_j+B)\bigr)$, the minimal number of translates of $B$ needed to cover $A$.”
  • [ABP2024-def-Hn]
    loc: arXiv PDF p.1, Abstract.
    quote: “Let $H_n$ be the minimal number such that any $n$-dimensional convex body can be covered by $H_n$ translates of interior of that body.”
  • [ABP2024-def-Hns]
    loc: arXiv PDF p.1, Abstract.
    quote: “Similarly $H_n^s$ is the corresponding quantity for symmetric bodies.”
  • [ABP2024-conj-Hn]
    loc: arXiv PDF p.1, Abstract.
    quote: “the famous Hadwiger’s covering conjecture (illumination conjecture) states that $H_n = H_n^s = 2^n$.”
  • [ABP2024-hadwiger-question]
    loc: arXiv PDF p.1, Introduction (paragraph after the definition of $H_n$).
    quote: “Hadwiger [17] raised the question of determining the value of $H_n = \min{C(K,\mathrm{int}(K)) : K \in K_n}$ for all $n \ge 3$.”
  • [ABP2024-lb-cube]
    loc: arXiv PDF p.1, Introduction (paragraph after the definition).
    quote: “Considering an $n$-cube, one immediately sees that $H_n \ge 2^n$,”
  • [ABP2024-ub-H3-16]
    loc: arXiv PDF p.3, Introduction (paragraph on low dimensions).
    quote: “then to $H_3 \le 16$ by Papadoperakis [24],”
  • [ABP2024-ub-H3-14]
    loc: arXiv PDF p.3, Introduction (paragraph on low dimensions).
    quote: “and then to $H_3 \le 14$ by Prymak [25].”
  • [ABP2024-H3s-8]
    loc: arXiv PDF p.3, Introduction (paragraph on the symmetric case).
    quote: “For the symmetric case, Lassak [20] obtained the sharp result $H_3^s = 8$,”

• [Had1957] Hadwiger, H. Ungelöste Probleme Nr. 20. Elemente der Mathematik 12(6) (1957), 121. Google Scholar. Publisher entry.

• [Pap1999] Papadoperakis, Ioannis. An estimate for the problem of illumination of the boundary of a convex body in $E^3$. Geometriae Dedicata 75(3) (1999), 275–285. DOI: 10.1023/A:1005056207406. Google Scholar.

• [Pry2023] Prymak, Andriy. A new bound for Hadwiger’s covering problem in $\mathbb{E}^3$. SIAM Journal on Discrete Mathematics 37(1) (2023), 17–24. DOI: 10.1137/22M1490314. Google Scholar. arXiv PDF.

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.