A set-valued extension of the Mazur–Ulam theorem
Volume 263 / 2022
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 the Hausdorff metric. We extend the Mazur–Ulam theorem in the following manner: The restriction $T|_X$ of a surjective isometry $T:\mathfrak {K}(X)\rightarrow \mathfrak {K}(Y)$ is an affine isometry from $X$ onto $Y$; if, in addition, one of $X$ and $Y$ is either strictly convex, or Gâteaux smooth, then $T(C)=\bigcup \{T|_X(x): x\in C\}$ for every $C\in \mathfrak {K}(X)$; and this is equivalent to “every surjective isometry $\mathfrak {K}(X)\rightarrow \mathfrak {K}(Y)$ is fully order preserving”.
