A Minkowski-type theorem for covering minima in the plane (Q1900039)

For \(K\) a plane convex body and \(L\) a lattice in \(E^2\), let \(\mu_i (K, L)\), \(i\in \{1; 2\}\), denote the covering minima given by \(\mu_i (K, L)= \min\{ t>0\): \(tK+ L\) meets every \((2-i)\)-dimensional affine subspace of \(E^2\}\). The author proves that \(\mu_1 (K, L)\cdot \mu_2 (K, L)\cdot V(K)= {3\over 4} \text{ det} (K)\), where \(V(K)\) denotes the area. In addition, it is shown that for each \(L\) there is, up to translations, dilatations and unimodular transformations, exactly one triangle, one parallelogram, one trapezoid, one pentagon and one hexagon such that equality holds.