Linearity of isometries between convex Jordan curves

Abstract: In this paper, we show that the $C^1$-differentiability of the norm of a two-dimensional normed space depends only on distances between points of the unit sphere in two different ways. As a consequence, we see that any isometry between the spheres of normed planes $\tau:S_X\to S_Y$ is linear, provided that there exist linearly independent $x,\overline{x}\in S_X$ where $S_X$ is not differentiable and that $S_X$ is piecewise differentiable. We end this work by showing that the isometry $\tau:C_X\to C_Y$ is linear even if it is not an isometry between spheres: every isometry between (planar) Jordan piecewise $C^1$-differentiable convex curves extends to $X$ whenever $X$ and $Y$ are strictly convex and the amount of non-differentiability points of $S_X$ and $S_Y$ is finite and greater than 2.