  Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if
the set of its vertices is S\cap L where S is a sphere having no lattice points
in its interior. D is called perfect if the only ellipsoid in R^n that contains
S\cap L is exactly S.
  For a vector v of the Leech lattice \Lambda_{24} we define \Lambda_{24}(v) to
be the lattice of vectors of \Lambda_{24} orthogonal to v. We studied Delaunay
polytopes of L=\Lambda_{24}(v) for |v|^2<=22. We found some remarkable examples
of Delaunay polytopes in such lattices and disproved a number of long standing
conjectures. In particular, we discovered:
  --Perfect Delaunay polytopes of lattice width 4; previously, the largest
known width was 2.
  --Perfect Delaunay polytopes in L, which can be extended to perfect Delaunay
polytopes in superlattices of L of the same dimension.
  --Polytopes that are perfect Delaunay with respect to two lattices $L\subset
L'$ of the same dimension.
  --Perfect Delaunay polytopes D for L with |Aut L|=6|Aut D|: all previously
known examples had |Aut L|=|Aut D| or |Aut L|=2|Aut D|.
  --Antisymmetric perfect Delaunay polytopes in L, which cannot be extended to
perfect (n+1)-dimensional centrally symmetric Delaunay polytopes.
  --Lattices, which have several orbits of non-isometric perfect Delaunay
polytopes.
  Finally, we derived an upper bound for the covering radius of
\Lambda_{24}(v)^{*}, which generalizes the Smith bound and we prove that it is
met only by \Lambda_{23}^{*}, the best known lattice covering in R^{23}.
