Blowing up convex sets in the plane (Q920403)

This nice article presents a solution to the following problem of Kannan and Lovász: If \(K\subset {\mathbb{R}}^ 2\) is a convex set such that each line of \({\mathbb{R}}^ 2\) meets \(K+{\mathbb{Z}}^ 2,\) does it follow that \(2K+{\mathbb{Z}}^ 2\) covers \({\mathbb{R}}^ 2?\) The author shows that in general \(\alpha K+{\mathbb{Z}}^ 2={\mathbb{R}}^ 2\) for \(\alpha \geq 1+(2/3)\sqrt{3}\approx 2.155,\) this bound is best possible and that the convex hull of (0,0), (1-\(\sqrt{3,1})\) is a worst-case example.