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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