Representation of surjective additive isometric embeddings between Hausdorff metric spaces of compact convex subsets in finite-dimensional Banach spaces

Suppose that X and Y are real finite-dimensional Banach spaces. Let (cc(X), H) be the metric space of all nonempty compact convex subsets of X equipped with the Hausdorff distance H, and let f : (cc(X), H) -> (cc(Y), H) be a surjective additive isometric embedding. Then there is a surjective linear isometric embedding (f) over bar : X -> Y such that f (A) = {(f) over bar (a) : a is an element of A} for every A is an element of cc(X).

Automated summary

In the context of real finite-dimensional Banach spaces X and Y, under certain conditions, there exists a specific linear isometric embedding that can embed X into Y.