# 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 $$b(C)le c$$.