  We generalize in positive characteristics some results of Bien and Brion on
log homogeneous compactifications of a homogeneous space under the action of a
connected reductive group. We also construct an explicit smooth log homogeneous
compactification of the general linear group by successive blow-ups starting
from a grassmannian. By taking fixed points of certain involutions on this
compactification, we obtain smooth log homogeneous compactifications of the
special orthogonal and the symplectic groups.
