Representation of surjective additive isometric embeddings between Hausdorff metric spaces of compact convex subsets in finite-dimensional Banach spaces
Volume 257 / 2021
Studia Mathematica 257 (2021), 111-119
MSC: Primary 46B04; Secondary 46B20.
DOI: 10.4064/sm200326-9-6
Published online: 23 July 2020
Abstract
Suppose that $X$ and $Y$ are real finite-dimensional Banach spaces. Let $(\operatorname{cc} (X),H)$ be the metric space of all nonempty compact convex subsets of $X$ equipped with the Hausdorff distance $H$, and let $f:(\operatorname{cc} (X),H)\rightarrow (\operatorname{cc} (Y),H)$ be a surjective additive isometric embedding. Then there is a surjective linear isometric embedding $\overline {f}:X\rightarrow Y$ such that $f(A)=\{\overline {f}(a): a\in A\}$ for every $A\in \operatorname{cc} (X)$.
