Any isometry between the spheres of absolutely smooth $2$-dimensional Banach spaces is linear

Abstract

We prove that any isometry between the unit spheres of $C^2$-smooth (more generally, absolutely smooth) smooth Banach spaces extends to a linear isometry of the Banach spaces. This answers the famous Tingley's problem in the class of absolutely smooth $2$-dimensional Banach spaces.

22 pages

References in corpus (2)

• Extension of isometries between unit spheres of finite-dimensional polyhedral Banach spaces
• A reflection on Tingley's problem and some applications

Cited by in corpus (3)

• Every 2-dimensional Banach space has the Mazur-Ulam property
• Linearity of isometries between convex Jordan curves
• Every non-smooth $2$-dimensional Banach space has the Mazur-Ulam property