标题： 随机几何图投影中无边交叉平面的存在性 显示英文标题

标题： Existence of a plane without edge crossings in projections of the random geometric graph

摘要： 考虑一个随机几何图$G$在一个凸体$W\subset \mathbb{R}^3$上，顶点集由强度为$t>0$的泊松点过程定义。 然后可以将$G$在平面上$L$绘制出来，方法是将所有顶点投影到$L$并在每对顶点之间存在边时用线段连接它们。 平面$L$的旋转会导致不同的绘制结果。 在本文中，我们证明，如果连接半径小于$\left(c^* \frac{\log t}{t^4}\right)^{1/8}$，则存在一个平面，使得投影中没有边交叉的概率趋于一，对于某些$c^*>0$。 显示英文摘要

摘要： Consider a random geometric graph $G$ on on a convex body $W\subset \mathbb{R}^3$ with a vertex set defined by a Poisson point process with intensity $t>0$. Then $G$ can be drawn on a plane $L$ by projecting all vertices onto $L$ and connecting each pair of vertices by a line segment whenever there is exists and edge between them. Rotations of the plane $L$ lead to different drawings. In this paper, we prove that the probability that there exists a plane such that there are no edge crossings in the projection tends to one if the connection radius is smaller than $\left(c^* \frac{\log t}{t^4}\right)^{1/8}$ for some $c^*>0$.