论文标题
球体Delaunay三角形的变形空间
The Deformation Space of Delaunay Triangulations of the Sphere
论文作者
论文摘要
在本文中，我们确定了单位$ 2 $ -SPHERE中刻有凸的Polyhedra的空间的拓扑，以及该单元的严格Delaunay Geodesic Triangulation的空间$ 2 $ -Sphere。这些空间可以被视为单位$ 2 $ -SPHERE的离散的差异群体。因此，自然可以猜想这些空间具有与光滑同行相同的同质拷贝类型。本文的主要结果证实了该单位$ 2 $ -sphere的猜想。从I. Rivin开发的三角形表面上的变异原理的观察结果。 相反，在扁平托里和凸多边形的情况下，类似的猜想不存在。我们将构造简单的扁平托里和凸多边形的示例，以便未连接Delaunay Geodesic三角形的相应空间。
In this paper, we determine the topology of the spaces of convex polyhedra inscribed in the unit $2$-sphere and the spaces of strictly Delaunay geodesic triangulations of the unit $2$-sphere. These spaces can be regarded as discretized groups of diffeomorphisms of the unit $2$-sphere. Hence, it is natural to conjecture that these spaces have the same homotopy types as those of their smooth counterparts. The main result of this paper confirms this conjecture for the unit $2$-sphere. It follows from an observation on the variational principles on triangulated surfaces developed by I. Rivin. On the contrary, the similar conjecture does not hold in the cases of flat tori and convex polygons. We will construct simple examples of flat tori and convex polygons such that the corresponding spaces of Delaunay geodesic triangulations are not connected.
