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Steinhaus Conjecture on minimizing the maximum isoperimetric ratio in planar tilings

Determine whether, for any tiling of the Euclidean plane in which every tile has diameter at least a fixed constant D > 0, the maximum isoperimetric ratio (perimeter(C)^2 / area(C)) among all tiles C is minimized by the regular hexagonal tiling.

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Background

The paper revisits a problem attributed to Steinhaus and listed as Problem C15 in Croft–Falconer–Guy’s Unsolved Problems in Geometry. It concerns whether the regular hexagonal tiling is optimal with respect to maximizing efficiency measured by the isoperimetric ratio among all tiles when a lower bound on tile diameter is imposed.

Using techniques developed in the paper, the authors provide partial results showing an average isoperimetric ratio minimization for convex, normal tilings and extensions in normed planes. However, the conjecture as originally stated (for all tilings with a diameter lower bound, minimizing the maximum ratio) is recalled as an open conjecture by Steinhaus.

References

We recall the following conjecture, appearing as Problem C15 in . For any tiling $\mathcal{T}$ in the Euclidean plane with tiles whose diameters are at least $D$ for some fixed $D > 0$, the maximum isoperimetric ratio $\frac{\perim(C)2}{\area(C)}$ of the cells $C$ of $\mathcal{T}$ is minimal if $\mathcal{T}$ is a regular hexagonal tiling.

— The Honeycomb Conjecture in normed planes and an alpha-convex variant of a theorem of Dowker (2406.10622 - Lángi et al., 15 Jun 2024) in Conjecture (Steinhaus), Section 6