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

TL;DR

This paper proves that any isometry between the unit spheres of absolutely smooth 2D Banach spaces extends to a linear isometry, solving Tingley's problem in this class.

Contribution

It establishes that isometries on the spheres of absolutely smooth 2D Banach spaces are necessarily linear, confirming Tingley's conjecture for this class.

Findings

Any isometry between the spheres extends linearly.

The result applies to all absolutely smooth 2D Banach spaces.

It solves Tingley's problem for this class.

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.