We bridge the two classical memoirs of G.F. Voronoi (1868--1908). In them he describes two reduction theory for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing.
By looking at the covering problem from a different perspective, the missing analogue is established. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. The punch line: All highly symmetric lattices give good packings, but, with the noble exception of the Leech lattice in dimension 24, they all give uneconomical coverings.
joint work with M. Dutour Sikiric and A. Schuermann