  Let G be a compact group acting on a Banach space L by means of linear
isometries. The action of G on L induces a natural continuous action on cc(L),
the hyperspace of all compact convex subsets of L endowed with the Hausdorff
metric topology. The main result of this paper states that the G-space cc(L) is
a G-AE. Under some extra assumptions, this result can be extended to CB(L), the
hyperspace of all closed and bounded convex subsets of L.
