Every non-smooth $2$-dimensional Banach space has the Mazur-Ulam property

TL;DR

This paper proves that all non-smooth two-dimensional Banach spaces possess the Mazur-Ulam property, meaning isometries between their unit spheres extend to linear isometries of the entire spaces.

Contribution

It establishes that every non-smooth 2D Banach space has the Mazur-Ulam property, filling a gap in the understanding of isometric extension properties.

Findings

All non-smooth 2D Banach spaces have the Mazur-Ulam property.

Isometries on the unit sphere extend to linear isometries.

The result applies specifically to non-smooth 2D Banach spaces.

Abstract

A Banach space $X$ has the $Mazur$-$Ulam$ $property$ if any isometry from the unit sphere of $X$ onto the unit sphere of any other Banach space $Y$ extends to a linear isometry of the Banach spaces $X,Y$. A Banach space $X$ is called $smooth$ if the unit ball has a unique supporting functional at each point of the unit sphere. We prove that each non-smooth 2-dimensional Banach space has the Mazur-Ulam property.