Flatness constant in dimension 3
Description of constant
A convex body $K\subset\mathbb{R}^d$ is hollow (lattice-free) with respect to a lattice $\Lambda$ if $\operatorname{int}(K)\cap\Lambda=\varnothing$. [CS2019-hollow-def]
For a hollow body, the lattice width is
\[w(K)\ :=\ \min_{u\in\mathbb{Z}^d\setminus\{0\}} \left(\max_{x\in K}u\cdot x-\min_{x\in K}u\cdot x\right).\]The flatness constant in dimension $d$ is
\[\mathrm{Flt}(d)\ :=\ \sup\{w(K): K\subset\mathbb{R}^d\ \text{hollow convex body}\},\]and is finite for each fixed $d$. [ACMS2021-Flt-def] [CS2019-flatness-def]
We define
\[C_{73}\ :=\ \mathrm{Flt}(3).\]An explicit hollow tetrahedron gives
\[\mathrm{Flt}(3)\ \ge\ 2+\sqrt{2}.\]An explicit published upper bound is
\[\mathrm{Flt}(3)\ <\ 3.972.\]Hence the best established range is
\[2+\sqrt{2}\ \le\ C_{73}\ <\ 3.972.\]Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $3.972$ | [ACMS2021] | Published explicit upper bound $\mathrm{Flt}(3)<3.972$. [ACMS2021-ub-3-972] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $1$ | Trivial lower bound for nonempty convex bodies. | |
| $2+\sqrt{2}$ | [CS2019] | Achieved by an explicit hollow non-lattice tetrahedron in dimension $3$. [CS2019-thm-1-1] |
Additional comments and links
-
A natural conjecture is $\mathrm{Flt}(3)=2+\sqrt{2}$. [CS2019-conjecture-1-2]
References
- [CS2019] Codenotti, Giulia; Santos, Francisco. Hollow polytopes of large width. Proceedings of the American Mathematical Society 148 (2019), no. 2, 835-850. DOI: https://doi.org/10.1090/proc/14721. arXiv PDF: https://arxiv.org/pdf/1812.00916.pdf. Google Scholar
- [CS2019-hollow-def] loc: arXiv PDF p.1, Introduction quote: “A convex body $K \subset \mathbb{R}^d$ is called hollow or lattice-free with respect to a lattice $\Lambda \simeq \mathbb{Z}^d$ if $\operatorname{int}(K)\cap \Lambda = \varnothing$.”
- [CS2019-flatness-def] loc: arXiv PDF p.1, Introduction quote: “The celebrated flatness theorem states that hollow bodies in fixed dimension $d$ have bounded lattice width. That is, for each fixed $d$, the supremum width among all hollow convex bodies in $\mathbb{R}^d$ is a certain constant $w_c(d)<\infty$.”
- [CS2019-thm-1-1] loc: arXiv PDF p.2, Theorem 1.1 quote: “There is a hollow (non-lattice) tetrahedron of width $2+\sqrt{2}$.”
- [CS2019-conjecture-1-2] loc: arXiv PDF p.2, Conjecture 1.2 quote: “Conjecture 1.2. The tetrahedron in Theorem 1.1 is the convex 3-body of largest width; that is, $w_c(3)=2+\sqrt{2}$.”
- [ACMS2021] Averkov, Gennadiy; Codenotti, Giulia; Macchia, Antonio; Santos, Francisco. A local maximizer for lattice width of 3-dimensional hollow bodies. Discrete Applied Mathematics 298 (2021), 129-142. DOI: https://doi.org/10.1016/j.dam.2021.04.009. arXiv PDF: https://arxiv.org/pdf/1907.06199.pdf. Google Scholar
- [ACMS2021-width-def] loc: arXiv PDF p.1, Introduction, paragraph with width definitions quote: “The width of $K$ in the direction of a linear functional $f$, denoted $\mathrm{width}(K,f)$, is the length of the segment $f(K)$. The (lattice) width of $K$ with respect to $\Lambda$, denoted $\mathrm{width}_{\Lambda}(K)$, is the minimum width with respect to all non-zero lattice functionals in $\vec{\Lambda}^{*}$, the linear lattice dual to $\vec{\Lambda}:=\Lambda-\Lambda$.”
- [ACMS2021-Flt-def] loc: arXiv PDF p.1, Introduction, paragraph containing Flatness Theorem statement quote: “The famous Flatness Theorem says that the so-called flatness constant, $\mathrm{Flt}(d):=\max{\mathrm{width}_{\Lambda}(K):\ K\subset\mathbb{R}^{d}\ \text{hollow with respect to}\ \Lambda}$, is a finite value.”
- [ACMS2021-ub-3-972] loc: arXiv PDF p.4, Introduction, bullet list of search-space restrictions quote: “• width bounded above by $3.972$, that is, we prove $\mathrm{Flt}(3)<3.972$;”
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.