A counter-example in geometry of numbers

An example of a symmetrical star domain S is given such that it does not contain any star domain which is minimal for coverings in the class of symmetrical star domains with covering constant 0(S). In the process it is also shown that the analogue of Farey's result for convex domains does not hold in the class of symmetrical star domains

Lattice double packings in the plane

It is proved that the lattice double packing density of a symmetrical convex domain is twice its lattice packing density. An example is given to show that this result cannot be extended to the class of symmetrical star domains