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

Taras Banakh

Published 2019-11-09Version 1

A $2$-dimensional Banach space $X$ is called $\breve W{}^{3,1}$-smooth if the polar parametrization $\mathbf p:\mathbb R\to S_X$ of its unit sphere $S_X=\{x\in X:\|x\|=1\}$ has locally absolutely continuous second derivative $\mathbf p''$ and the vector $\mathbf p''(t)$ is not collinear to $\mathbf p'(t)$ for almost all $t\in\mathbb R$. We prove that any isometry $f:S_X\to S_Y$ between the unit spheres of $S\breve W{}^{3,1}$-smooth $2$-dimensional Banach spaces $X,Y$ extends to a linear isometry $\bar f:X\to Y$ of the Banach spaces $X,Y$. This answers the famous Tingley's problem in the class of $\breve W{}^{3,1}$-smooth $2$-dimensional Banach spaces.