A set-valued extension of the Mazur–Ulam theorem

Lixin Cheng, Zheming Zheng Studia Mathematica MSC: Primary 46B04; Secondary 46B20. DOI: 10.4064/sm200120-9-2 Opublikowany online: 21 October 2021

Streszczenie

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”.

Autorzy

  • Lixin ChengSchool of Mathematical Sciences
    Xiamen University
    Xiamen, 361005, P.R. China
    e-mail
  • Zheming ZhengSchool of Mathematical Sciences
    Xiamen University
    Xiamen, 361005, P.R. China
    e-mail

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