Flatness constant in dimension 3 ($C_{73}$)

• A natural conjecture is $\mathrm{Flt}(3)=2+\sqrt{2}$. [CS2019-conjecture-1-2]

• Background page on geometry of numbers

References (from the repository page)

• [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 $\mathrm{Flt}(3)<3.972$;"

Repository table (commit 2c1968c, 26 September 2026): best known lower bound $2+\sqrt{2}$; best known upper bound $<3.972$. Values marked with an asterisk have minimal verification.

Known upper bounds

Known lower bounds