A convex set-valued version of the Mazur-Ulam theorem on Asplund spaces

Abstract

For a Banach space X, let C(X) be the cone consisting of all nonempty bounded closed convex subsets of X endowed with the Hausdorff metric. In this paper, we show that if one of the two Banach spaces X and Y is an Asplund space, then for every surjective isometry f:C(X)→C(Y), the restriction f|X is a surjective affine isometry from X to Y. If, in addition, one of X and Y is Fréchet smooth, or, locally uniformly convex, then f(C)=⋃{f(x):x∈C} for all C∈C(X).