A reflection on Tingley's problem and some applications

Abstract

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.

28 pages, one figure, accepted for publication in the Journal of Mathematical Analysis and Applications

References in corpus (5)

• The geometry of Minkowski spaces -- a survey. Part I
• Extension of isometries between unit spheres of finite-dimensional polyhedral Banach spaces
• Tingley's problem through the facial structure of operator algebras
• Mankiewicz's theorem and the Mazur--Ulam property for C*-algebras
• A characterisation of Euclidean normed planes via bisectors

Cited by in corpus (9)

• Extension of isometries from the unit sphere of a rank-2 Cartan factor
• Every 2-dimensional Banach space has the Mazur-Ulam property
• Any isometry between the spheres of absolutely smooth $2$-dimensional Banach spaces is linear
• Metric characterisation of unitaries in JB${}^{\ast }$-algebras
• Every non-smooth $2$-dimensional Banach space has the Mazur-Ulam property
• Linearity of isometries between convex Jordan curves
• Phase-isometries on the unit sphere of $C(K)$
• A property in vector-valued function spaces
• The Mazur--Ulam property in ${\ell }_{\infty }$-sum and $c_0$-sum of strictly convex Banach spaces