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

Volume 257 / 2021

Yu Zhou, Zihou Zhang, Chunyan Liu 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 $A\in \operatorname{cc} (X)$.

Authors

  • Yu ZhouSchool of Mathematics, Physics and Statistics
    Shanghai University of Engineering Science
    Shanghai, 201620, P.R. China
    e-mail
  • Zihou ZhangSchool of Mathematics, Physics and Statistics
    Shanghai University of Engineering Science
    Shanghai, 201620, P.R. China
    e-mail
  • Chunyan LiuSchool of Mathematics, Physics and Statistics
    Shanghai University of Engineering Science
    Shanghai, 201620, P.R. China
    e-mail

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