On the Interior \(H\)-points of \(H\)-polygons

Abstract

An \(H\)-polygon is a simple polygon whose vertices are \(H\)-points, which are points of the set of vertices of a tiling of \(\mathbb{R}^2\) by regular hexagons of unit edge. Let \(G(v)\) denote the least possible number of \(H\)-points in the interior of a convex \(H\)-polygon \(K\) with \(v\) vertices. In this paper, we prove that \(G(8) = 2\), \(G(9) = 4\), \(G(10) = 6\), and \(G(v) \geq \lceil \frac{v^2}{16\pi^2}-\frac{v}{4}+\frac{1}{2}\rceil – 1\) for all \(v \geq 11\), where \(\lceil x \rceil\) denotes the minimal integer more than or equal to \(x\).