论文标题
径向膨胀可保留双曲线凸度和径向收缩可保留球形凸度
Radial expansion preserves hyperbolic convexity and radial contraction preserves spherical convexity
论文作者
论文摘要
在平面上，通过径向膨胀和集合的收缩来保留集合的凸度，并且在其内部的任何点上都可以保留。使用双曲线几何形状的Poincaré磁盘模型，我们证明双曲线凸的径向膨胀围绕其中一个点始终保持双曲线凸度。使用球形的立体投影，我们证明球形凸的径向收缩围绕它内部的一个点，因此初始集合包含在以此为中心的闭合半球中，总是保留球形凸度。
On a flat plane, convexity of a set is preserved by both radial expansion and contraction of the set about any point inside it. Using the Poincaré disk model of hyperbolic geometry, we prove that radial expansion of a hyperbolic convex set about a point inside it always preserves hyperbolic convexity. Using stereographic projection of a sphere, we prove that radial contraction of a spherical convex set about a point inside it, such that the initial set is contained in the closed hemisphere centred at that point, always preserves spherical convexity.
