A set-valued extension of the Mazur–Ulam theorem

Abstract

The Mazur–Ulam theorem states that every surjective isometry from a Banach space $X$ to a Banach space $Y$ is necessarily affine. Let $\mathfrak {K}(X)$ (resp. $\mathfrak {K}(Y)$) be the cone of all compact convex subsets of $X$ (resp. $Y$) endowed with t