Hadwiger covering / illumination number in 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. <a href="#ABP2024-equivalence-illumination">[ABP2024-equivalence-illumination]</a>

Given sets $K, L \subset R^{n}$ , let $C (K, L)$ be the minimal number of translates of $L$ needed to cover $K$ . <a href="#ABP2024-def-CKL">[ABP2024-def-CKL]</a>

For a convex body $K \subset R^{n}$ , write $int (K)$ 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. <a href="#ABP2024-def-Hn">[ABP2024-def-Hn]</a>

The constant of interest here is $H_{3}$ . <a href="#ABP2024-def-Hn">[ABP2024-def-Hn]</a>

For symmetric convex bodies one also considers the symmetric covering number $H_{n}^{s}$ , defined analogously. <a href="#ABP2024-def-Hns">[ABP2024-def-Hns]</a>

Known upper bounds

Bound	Reference	Comments
$16$	<a href="#Pap1999">[Pap1999]</a>	Previous best bound: $H_{3} \leq 16$ (Papadoperakis). <a href="#ABP2024-ub-H3-16">[ABP2024-ub-H3-16]</a>
$14$	<a href="#Pry2023">[Pry2023]</a>	Best known general upper bound: $H_{3} \leq 14$ (attributed to Prymak). <a href="#ABP2024-ub-H3-14">[ABP2024-ub-H3-14]</a>

Known lower bounds

Bound	Reference	Comments
$8$	Classical (cube)	$H_{3} \geq 2^{3} = 8$ (already forced by the cube / parallelotope). <a href="#ABP2024-lb-cube">[ABP2024-lb-cube]</a>

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 $H_{3} = 8$ . <a href="#ABP2024-conj-Hn">[ABP2024-conj-Hn]</a>

Origin of the conjecture. Hadwiger posed the covering problem in 1957. <a href="#ABP2024-hadwiger-question">[ABP2024-hadwiger-question]</a> <a href="#Had1957">[Had1957]</a>

Centrally symmetric case in dimension $3$ . The symmetric variant is known exactly: $H_{3}^{s} = 8$ (and is sharp). <a href="#ABP2024-H3s-8">[ABP2024-H3s-8]</a>

Surveys/background for the general illumination/covering problem include <a href="#ABP2024">[ABP2024]</a>.

References

<a id="ABP2024"></a>[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.
<a id="ABP2024-equivalence-illumination"></a>[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,”
<a id="ABP2024-def-CKL"></a>[ABP2024-def-CKL] loc: arXiv PDF p.1, Introduction (definitions paragraph). quote: “we denote by $C (A, B) := min (N : \exists t_{1}, \dots, t_{N} \in E^{n} satisfying A \subset ⋃_{j = 1}^{N} (t_{j} + B))$ , the minimal number of translates of $B$ needed to cover $A$ .”
<a id="ABP2024-def-Hn"></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.”
<a id="ABP2024-def-Hns"></a>[ABP2024-def-Hns] loc: arXiv PDF p.1, Abstract. quote: “Similarly $H_{n}^{s}$ is the corresponding quantity for symmetric bodies.”
<a id="ABP2024-conj-Hn"></a>[ABP2024-conj-Hn] loc: arXiv PDF p.1, Abstract. quote: “the famous Hadwiger’s covering conjecture (illumination conjecture) states that $H_{n} = H_{n}^{s} = 2^{n}$ .”
<a id="ABP2024-hadwiger-question"></a>[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 $H_{n} = min C (K, int (K)) : K \in K_{n}$ for all $n \geq 3$ .”
<a id="ABP2024-lb-cube"></a>[ABP2024-lb-cube] loc: arXiv PDF p.1, Introduction (paragraph after the definition). quote: “Considering an $n$ -cube, one immediately sees that $H_{n} \geq 2^{n}$ ,”
<a id="ABP2024-ub-H3-16"></a>[ABP2024-ub-H3-16] loc: arXiv PDF p.3, Introduction (paragraph on low dimensions). quote: “then to $H_{3} \leq 16$ by Papadoperakis [24],”
<a id="ABP2024-ub-H3-14"></a>[ABP2024-ub-H3-14] loc: arXiv PDF p.3, Introduction (paragraph on low dimensions). quote: “and then to $H_{3} \leq 14$ by Prymak [25].”
<a id="ABP2024-H3s-8"></a>[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 $H_{3}^{s} = 8$ ,”

<a id="Had1957"></a>[Had1957] Hadwiger, H. Ungelöste Probleme Nr. 20. Elemente der Mathematik 12(6) (1957), 121. Google Scholar. Publisher entry.

<a id="Pap1999"></a>[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.

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

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.