论文标题
刻在2个球体中的随机多型的美丽
The Beauty of Random Polytopes Inscribed in the 2-sphere
论文作者
论文摘要
考虑在$ \ mathbb {r}^d $中的单位球上的一组随机点，可以均匀地采样或泊松点过程。它的凸壳是一个随机铭刻的多型，其边界近似球体。我们专注于$ d = 3 $的情况，其中有基本的证明和引人入胜的公式用于公制属性。特别是，我们研究了急性面的比例，预期的固有体积，总边长以及到固定点的距离。最后，我们将结果概括为椭圆形密度。
Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case $d=3$, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.
