  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.
