co.combinatorics – Existence of a honeycomb composed by nearly-hyperspherical $d$-dimensional cells having the same shape and size


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 let $b(C)$ be equal to $frac{ell(C)}{s(C)}$ (informally, we view $b(C)$ as a measure providing information about to what extent $C$ is similar to a $d$-dimensional ball).


Example: For d=2 we have if we consider the hexagonal tiling $hinmathcal{H}$, $C$ 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 $b(C)=frac{2}{sqrt{3}}$. For $d=3$, we could consider the honeycomb $hinmathcal{H}$ made up of truncated octahedrons and calculate $b(C)$. Finally, in general, if we consider hypercubic honeycomb $hinmathcal{H}$, we have $b(C)=sqrt{d}$, 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?

For any $d>1$ there is a constant $cinmathbb{R}$ and a $d$-dimensional honeycomb $hinmathcal{H}$ such that $b(C)le c$.