A reflection on Tingley's problem and some applications

In this paper we show how some metric properties of the unit sphere of a normed space can help to approach a solution to Tingley's problem. In our main result we show that if an onto isometry between the spheres of strictly convex spaces is the identity when restricted to some relative open sub...

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Bibliographic Details
Published in:arXiv.org 2024-02
Main Author: Javier Cabello Sánchez
Format: Article
Language:English
Subjects:
Curvature
Online Access:Get full text
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Summary:In this paper we show how some metric properties of the unit sphere of a normed space can help to approach a solution to Tingley's problem. In our main result we show that if an onto isometry between the spheres of strictly convex spaces is the identity when restricted to some relative open subset, then it is the identity. This implies that an onto isometry between the unit spheres of strictly convex finite dimensional spaces is linear if and only it is linear on a relative open set. We prove the same for arbitrary two-dimensional spaces and obtain that every two-dimensional, non strictly convex, normed space has the Mazur-Ulam Property. We also include some other less general, yet interesting, results, along with a generalisation of curvature to normed spaces.
ISSN:2331-8422
DOI:10.48550/arxiv.1810.02461