# 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, 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 $$b(h)\le c$$.

• "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 $d$)? – fedja Aug 25 '20 at 19:06
• 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. – fedja Aug 25 '20 at 22:59
• 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. – Steve Huntsman Aug 26 '20 at 13:33
• 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) – Yoav Kallus Aug 26 '20 at 16:09
• 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. – Adam P. Goucher Aug 27 '20 at 11:35