Several point symmetrizations of a convex curve $\Gamma$ are introduced and one, the affinely invariant ‘central symmetric transform’ (CST) with respect to a given basepoint inside $\Gamma$, is investigated in detail. Examples for $\Gamma$ include triangles, rounded triangles, ellipses, curves defined by support functions and piecewise smooth curves. Of particular interest is the region of basepoints for which the CST is convex (this region can be empty but its complement in the interior of $\Gamma$ is never empty). The (local) boundary of this region can have cusps and in principle it can be determined from a geometrical construction for the tangent direction to the CST.

中文翻译：

---

凸平面曲线的对称化

引入了凸曲线$ \ Gamma $的几种点对称性，其中一种是针对$ \ Gamma $内部给定基点的仿射不变的“中心对称变换”（CST）。$ \ Gamma $的示例包括三角形，圆角三角形，椭圆形，由支撑函数定义的曲线和分段平滑曲线。特别令人感兴趣的是CST凸出的基点区域（该区域可以为空，但其在$ \ Gamma $内部的补数永远不会为空）。该区域的（局部）边界可能具有尖点，并且原则上可以从与CST相切的方向的几何构造确定。