  Let G be a connected real reductive Lie group acting linearly on a finite
dimensional vector space V over R. This action admits a Kempf-Ness function and
so we have an associated gradient map. If G is Abelian we explicitly compute
the image of G orbits under the gradient map, generalizing a result proved by
Kac and Peterson. A similar result is proved for the gradient map associated to
the natural $G$ action on P(V). We also investigate the convex hull of the
image of the gradient map restricted on the closure of G orbits. Finally, we
give a new proof of the Hilbert-Mumford criterion for real reductive Lie groups
avoiding any algebraic result
