Equivariant absolute extensor property on hyperspaces of convex sets

Let G be a compact group acting on a Banach space L by means of affine transformations. The action of G on L   induces a natural continuous action on cc(L)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)cc(L) is a G  -AE. Under some extra assumptions, this result can be extended to CB(L)CB(L), the hyperspace of all closed and bounded convex subsets of L.