A covering problem for plane lattices (Q1196147)

The author considers convex covering sets with respect to a plane lattice \(L\) having parallelograms as fundamental cells. (A plane convex body is called a covering set of \(L\) if it contains, in each position, at least one point of \(L\).) A basis \(\{b_ 1,b_ 2\}\) of \(L\) is said to be reduced if three conditions are satisfied: 1) \(b_ 1\in\{v\in L\backslash\{0\}: \| v\|\) is minimal\} 2) \(b_ 2\in\{v\in L\backslash\{0\}: b_ 1\), \(v\) are a basis of \(L\), \(\| v\|\) is minimal\} 3) \(b_ 1\cdot b_ 2\geq 0\). For a reduced basis \(\{b_ 1,b_ 2\}\) of \(L\) with \(b_ 1=(a,0)\), \(b_ 2=(\cos\varphi, \sin\varphi)\), \(0<a\leq 1\), and \(0\leq\varphi\leq\pi/2\) it is shown: If the minimal width (thickness) \(\Delta(K)\) of a plane convex body \(K\) satisfies \(\Delta(K)\geq\sin\varphi+a\sqrt{{3\over 2}}\), then \(K\) is a covering set. The case of equality is necessary precisely for certain regular triangles. Thus, the author continues investigations of \textit{P. R. Scott} [Mathematika 20, 247-252 (1973; Zbl 0287.52002)] with respect to the integral lattice \(\mathbb{Z}^ 2\).