The polygon valued Minkowski valuations in ${\mathbb{R}}^2$ intertwining with the special linear group are completely classified in this paper without any additional assumptions. Our results characterize projection bodies and difference bodies in ${\mathbb{R}}^2$. There are also some $\mathrm{SL}(2)$ intertwining Minkowski valuations that can only be defined in ${\mathbb{R}}^2$. It somehow suggests why such classification problems are more complicated than higher dimensional spaces.

中文翻译：

---

SL(2) 在平面上交织 Minkowski 估值

多边形估值 Minkowski 估值 ${\mathbb{电阻}}^2$与特殊线性群交织在本文中完全分类，没有任何额外的假设。我们的结果表征了投影体和差异体${\mathbb{电阻}}^2$. 还有一些$\mathrm{SL}(2)$ 交织的 Minkowski 估值只能在 ${\mathbb{电阻}}^2$. 它以某种方式暗示了为什么此类分类问题比高维空间更复杂。