The Honeycomb Conjecture in normed planes and an alpha-convex variant of a theorem of Dowker

TL;DR

This paper extends the Honeycomb Conjecture to normed planes, showing minimal average squared perimeter for hexagonal tilings and exploring related geometric inequalities and their implications.

Contribution

It introduces a new minimality result for convex tilings in normed planes and connects it to an alpha-convex variant of Dowker's theorem, expanding the conjecture's scope.

Findings

Minimal average squared perimeter for convex tilings in normed planes occurs with hexagonal tilings.

Established a link between perimeter minimization and an alpha-convex variant of Dowker's theorem.

Provided partial answers to Steinhaus's problem on isoperimetric ratios of tiling cells.

Abstract

The Honeycomb Conjecture states that among tilings with unit area cells in the Euclidean plane, the average perimeter of a cell is minimal for a regular hexagonal tiling. This conjecture was proved by L. Fejes T\'oth for convex tilings, and by Hales for not necessarily convex tilings. In this paper we investigate the same question for tilings of a given normed plane, and show that among normal, convex tilings in a normed plane, the average squared perimeter of a cell is minimal for a tiling whose cells are translates of a centrally symmetric hexagon. We also show that the question whether the same statement is true for the average perimeter of a cell is closely related to an $\alpha$-convex variant of a theorem of Dowker on the area of polygons circumscribed about a convex disk. Exploring this connection we find families of norms in which the average perimeter of a cell of a tiling is minimal for a hexagonal tiling, and prove some additonal related results. Finally, we apply our method to give a partial answer to a problem of Steinhaus about the isoperimetric ratios of cells of certain tilings in the Euclidean plane, appeared in an open problem book of Croft, Falconer and Guy.