Every 2-dimensional Banach space has the Mazur-Ulam property

TL;DR

This paper proves that in 2-dimensional Banach spaces, any isometry between their unit spheres extends to a linear isometry, solving Tingley's problem for this class.

Contribution

It establishes that all 2-dimensional Banach spaces satisfy the Mazur-Ulam property, confirming Tingley's conjecture in this setting.

Findings

Every isometry between unit spheres extends linearly.

Tingley's problem is affirmatively solved for 2D Banach spaces.

The result applies universally to all 2D Banach spaces.

Abstract

We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear isometry of the Banach spaces. This resolves the famous Tingley's problem in the class of 2-dimensional Banach spaces.