Linearity of isometries between convex Jordan curves

In this paper, we show that the C1-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 τ:SX→SY is linear, provided that there exist linearly independent x,x‾∈SX where SX is not differentiable and that SX is piecewise differentiable. We end this work by showing that the isometry τ:CX→CY is linear even if it is not an isometry between spheres: every isometry between (planar) Jordan piecewise C1-differentiable convex curves extends to X whenever X and Y are strictly convex and the amount of non-differentiability points of SX and SY is finite and greater than 2.