凸Jordan曲线间等距线的线性
Linearity of isometries between convex Jordan curves
摘要：
本文证明了二维赋范空间的范数的$C^1$ -可微性仅以两种不同的方式依赖于单位球面上点之间的距离。因此，我们看到，任何等距的领域之间的赋范平面$τ:S_X\to S_Y$是线性的，只要存在线性无关的$x,\overline{x}\in S_X$，其中$S_X$是不可微的，而$S_X$是分段可微的。我们结束这项工作表明，等距$τ:C_X\to C_Y$是线性的，即使它不是一个等距之间的领域：（平面）约旦分段$C^1$ -可微凸曲线之间的每一个等距延伸到$X$时，$X$和$Y$是严格凸的和数量的不可微点的$S_X$和$S_Y$是有限的，大于2。
英文摘要：
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 $τ:S_X\to S_Y$ 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 $τ:C_X\to C_Y$ 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.
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