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On the Extension Problems of Isometric and Nonexpansive Mappings

In: Mathematics Without Boundaries

Author

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  • Xiuzhong Yang

    (Hebei Normal University, and Hebei Key Laboratory of Computational Mathematics and Applications, College of Mathematics and Information Science)

  • Xiaopeng Zhao

    (Zhejiang University, Department of Mathematics)

Abstract

We present a survey of ideas and results stemming from the Tingley’s problem: Let X and Y be real normed spaces with unit spheres $S_1(X)$ and $S_1(Y)$ , respectively. Suppose that $V_0:S_1(X)\rightarrow S_1(Y)$ is a surjective isometry. Is V 0 necessarily the restriction to $S_1(X)$ of a linear or affine isometry on X? Two more related isometric extension problems for nonsurjective isometry or nonexpansive mapping are also proposed and studied. A series of important results are outlined and some open problems are raised.

Suggested Citation

  • Xiuzhong Yang & Xiaopeng Zhao, 2014. "On the Extension Problems of Isometric and Nonexpansive Mappings," Springer Books, in: Themistocles M. Rassias & Panos M. Pardalos (ed.), Mathematics Without Boundaries, edition 127, pages 725-748, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4939-1106-6_24
    DOI: 10.1007/978-1-4939-1106-6_24
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