  G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two
reduction theories 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.
In this paper we bridge the two classical memoirs.
  By looking at the covering problem from a different perspective, we discover
the missing analogue. Instead of trying to find lattices giving economical
coverings we consider lattices giving, at least locally, very uneconomical
ones. We classify local covering maxima up to dimension 6 and prove their
existence in all dimensions beyond.
  New phenomena arise: Many highly symmetric lattices turn out to give
uneconomical coverings; the covering density function is not a topological
Morse function. Both phenomena are in sharp contrast to the packing problem.
