Add Hadwiger covering number.

Author: damek

README.md

@@ -6,7 +6,7 @@ A curated collection of optimization constants $C$ in mathematics, often arising
 
 ## Table of Constants
 
-We are arbitrarily numbering the constants as $C\_{1}$, $C\_{2}$, etc., mostly based on the order in which the constants were added to the repository.  Constants that are in a family of similar constants will also be given letter suffixes (e.g. $C\_{1a}$, $C\_{1b}$).
+We are arbitrarily numbering the constants as $C_{1}$, $C_{2}$, etc., mostly based on the order in which the constants were added to the repository.  Constants that are in a family of similar constants will also be given letter suffixes (e.g. $C_{1a}$, $C_{1b}$).
 
 | Number | Description | Best lower bound | Best upper bound |
 | ------ | ----------- | ---------------- | ---------------- |
@@ -58,6 +58,7 @@ We are arbitrarily numbering the constants as $C\_{1}$, $C\_{2}$, etc., mostly b
 | [36](https://teorth.github.io/optimizationproblems/constants/36a.html) | Sphere packing density in $\mathbb{R}^4$ | $\pi^2/16 \approx 0.616850$ | 0.644421 |
 | [37](https://teorth.github.io/optimizationproblems/constants/37a.html) | The degree--sensitivity exponent | $\log_{3}(6) \approx 1.63093$ | 2 |
 | [38](https://teorth.github.io/optimizationproblems/constants/38a.html) | Square-lattice self-avoiding walk connective constant | 2.625622 | 2.679193 |
+| [39](https://teorth.github.io/optimizationproblems/constants/39a.html) | Hadwiger covering / illumination number in $\mathbb{R}^3$ | 8 | 14 |
 
 
 ## Recent progress

constants/39a.md

@@ -0,0 +1,88 @@
+# 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.
+<a href="#ABP2024-equivalence-illumination">[ABP2024-equivalence-illumination]</a>
+
+Given sets $K,L\subset \mathbb{R}^n$, 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 $K\subset \mathbb{R}^n$, 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.
+<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: $H_3 \le 16$ (Papadoperakis). <a href="#ABP2024-ub-H3-16">[ABP2024-ub-H3-16]</a> |
+| $14$ | <a href="#Pry2023">[Pry2023]</a> | Best known general upper bound: $H_3 \le 14$ (attributed to Prymak). <a href="#ABP2024-ub-H3-14">[ABP2024-ub-H3-14]</a> |
+
+## Known lower bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| $8$ | Classical (cube) | $H_3 \ge 2^3 = 8$ (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](https://doi.org/10.4153/S0008439525000384). [Google Scholar](https://scholar.google.com/scholar?q=On+Hadwiger%E2%80%99s+covering+problem+in+small+dimensions+Arman+Bondarenko+Prymak). [arXiv PDF](https://arxiv.org/pdf/2404.00547.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 $C(A,B):=\min\bigl(N:\exists t_1,\dots,t_N\in\mathbb{E}^n\text{ satisfying }A\subset\bigcup_{j=1}^N(t_j+B)\bigr)$, the minimal number of translates of $B$ needed to cover $A$.”
+	- <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 $H_n = \min\{C(K,\mathrm{int}(K)) : K \in K_n\}$ for all $n \ge 3$.”
+	- <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 $H_n \ge 2^n$,”
+	- <a id="ABP2024-ub-H3-16"></a>**[ABP2024-ub-H3-16]**  
+	  **loc:** arXiv PDF p.3, Introduction (paragraph on low dimensions).  
+	  **quote:** “then to $H_3 \le 16$ 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 $H_3 \le 14$ 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](https://scholar.google.com/scholar?q=Ungel%C3%B6ste+Probleme+Nr.+20+Hadwiger+1957+Elemente+der+Mathematik). [Publisher entry](https://www.e-periodica.ch/cntmng?pid=edm-001%3A1957%3A12%3A%3A246).
+
+- <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](https://doi.org/10.1023/A:1005056207406). [Google Scholar](https://scholar.google.com/scholar?q=An+estimate+for+the+problem+of+illumination+of+the+boundary+of+a+convex+body+in+E%5E3+Papadoperakis+1999).
+
+- <a id="Pry2023"></a>**[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](https://doi.org/10.1137/22M1490314). [Google Scholar](https://scholar.google.com/scholar?q=A+new+bound+for+Hadwiger%27s+covering+problem+in+E%5E3+Prymak+2023). [arXiv PDF](https://arxiv.org/pdf/2112.10698).
+
+## Contribution notes
+
+Prepared with assistance from ChatGPT 5.2 Pro.