On Coverings by Minkowski Balls in the Plane and a Duality

Lattice coverings in the real plane by Minkowski balls are studied. We exploit the duality of admissible lattices of Minkowski balls and inscribed convex symmetric hexagons of these balls. An explicit moduli space of the areas of these hexagons is constructed, giving the values of the determinants of corresponding covering lattices. Low and upper bounds for covering constants of Minkowski balls are given. The best known value of covering density of Minkowski ball is obtained. One conjecture and one problem are formulated.