  We study weighted Sobolev inequalities on open convex cones endowed with
$\alpha$-homogeneous weights satisfying a certain concavity condition. We
establish a so-called reduction principle for these inequalities and
characterize optimal rearrangement-invariant function spaces for these weighted
Sobolev inequalities. Both optimal target and optimal domain spaces are
characterized. Abstract results are accompanied by general yet concrete
examples of optimal function spaces. For these examples, the class of so-called
Lorentz--Karamata spaces, which contains in particular Lebesgue spaces, Lorentz
spaces, and some Orlicz spaces, is used.
