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

The statement

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).

[ACMS2021-width-def]

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}.

[CS2019-thm-1-1]

An explicit published upper bound is

\mathrm{Flt}(3)\ <\ 3.972.

[ACMS2021-ub-3-972]

Hence the best established range is

2+\sqrt{2}\ \le\ C_{73}\ <\ 3.972.

What gets accepted

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