Let $\mathcal{H}$ the class of all honeycombs composed by $d$-dimensional cells $C$ having all the same shape and size in a $d$-dimensional space $\mathcal{S}$. Let $s(C)$ and $\ell(C)$ be respectively the length of the smallest and largest segment obtained through an orthogonal projection of $C$ onto a straight line over all straight lines in $\mathcal{S}$. Finally, for any given $h\in\mathcal{H}$, let $b(h)$ be equal to $\frac{\ell(C)}{s(C)}$ (informally, we view $b(h)$ as a measure providing information about to what extent the cells $C$ of $h$ are similar to a $d$-dimensional ball).

Example: For d=2 we have if we consider the hexagonal tiling $h\in\mathcal{H}$, $C$ is the hexagon, the radius of the circumscribed circle is equal to $\frac{2}{\sqrt{3}}$ times the apothem. Hence, it is immediate to verify that we have $b(h)=\frac{2}{\sqrt{3}}$. For $d=3$, we could consider the honeycomb $h\in\mathcal{H}$ made up of truncated octahedrons and calculate $b(h)$. Finally, in general, if we consider $d$-dimensional hypercubic honeycomb $h\in\mathcal{H}$, we have $b(h)=\sqrt{d}$, showing that this honeycomb is far from being composed by nearly-hyperspherical $d$-dimensional cells.

Question: How can we prove or disprove the following conjecture?

There exists a constant $c\in\mathbb{R}$ (that does not depend on $d$) such that, for all $d>1$, we have a $d$-dimensional honeycomb $h\in\mathcal{H}$ for which $b(h)\le c$.

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    $\begingroup$ "For any $d>1$ there exists a constant $c\in\mathbb R$" Are you sure that you wanted this order of quantifiers and not the reverse one (i.e., that $c$ is allowed to depend on $d$)? $\endgroup$ – fedja Aug 25 '20 at 19:06
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    $\begingroup$ Apparently Rogers showed back in 1950 that every packing lattice for a unit ball is contained in a 3-covering packing lattice (i.e., the new lattice is still packing but if you extend the balls 3 times, then they will cover the entire space). This would give the estimate $c\le 3$ (just consider the Voronoi cells for that lattice). The standard reference is C. A. Rogers. A note on coverings and packings.J. London Math. Soc., 25:327–331, 1950 but I could not check it myself because it is behind a paywall. $\endgroup$ – fedja Aug 25 '20 at 22:59
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    $\begingroup$ Permutohedra would be a good case to investigate here (generalizing hexagons and truncated octahedra) but I couldn't easily find the ratio of in/circumradii anywhere obvious and was too lazy to compute it. Exercise for the reader. $\endgroup$ – Steve Huntsman Aug 26 '20 at 13:33
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    $\begingroup$ To add to Fedja's comment, the problem of finding a lattice with the smallest possible ratio of covering radius to packing radius is also the subject of a paper of Schurmann and Vallentin (arxiv.org/abs/math/0403272), where they cite a result of Butler for the statement that the optimal ratio is asymptotically 2+o(1) $\endgroup$ – Yoav Kallus Aug 26 '20 at 16:09
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    $\begingroup$ Butler's result is an upper bound, that the packing-covering ratio is $\leq 2 + o(1)$. I don't think that an analogous lower bound has been established. $\endgroup$ – Adam P. Goucher Aug 27 '20 at 11:35

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