Linearity of isometries between convex Jordan curves

TL;DR

This paper investigates the conditions under which isometries between convex Jordan curves and spheres in two-dimensional normed spaces are linear, revealing new links between differentiability, convexity, and isometric mappings.

Contribution

It establishes that isometries between certain convex Jordan curves and spheres are linear under specific differentiability and convexity conditions, extending previous results.

Findings

Isometries between spheres are linear if the sphere is piecewise differentiable and not fully differentiable.

Any isometry between planar convex Jordan curves extends to a linear isometry under certain differentiability and convexity conditions.

Linearity of isometries holds even when they are not sphere isometries but between convex Jordan curves with finite non-differentiability points.

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.