论文标题
楔子上随机点的球形凸壳
Spherical convex hull of random points on a wedge
论文作者
论文摘要
考虑两个半空间$ h_1^+$和$ h_2^+$ in $ \ mathbb {r}^{d+1} $，其边界hyperplanes $ h_1 $和$ h_2 $是正交的，并通过原点。交点$ \ MATHBB {s} _ {2，+}^d：= \ Mathbb {s}^d \ cap H_1^+cap H_1^+cap H_2^+$是$ d $ d $ -dimensional sphere sphere $ \ mathbb {s} $ dist $ sphemine的球形凸子集楔子。在$ \ mathbb {s} _ {2，+}^d $上随机选择$ n $独立的随机点，并考虑这些点的球形凸形船体的预期方面数。结果表明，直到低阶的条款，这种期望就会像$ \ log n $的常数倍数一样增长。对于$ \ mathbb {s} _ {2，+}^d $，在均质泊松点进程的预期方面数量也获得了类似的行为。将结果与经典的欧几里得随机多型和半球上的球形随机多型的相应行为进行了比较。
Consider two half-spaces $H_1^+$ and $H_2^+$ in $\mathbb{R}^{d+1}$ whose bounding hyperplanes $H_1$ and $H_2$ are orthogonal and pass through the origin. The intersection $\mathbb{S}_{2,+}^d:=\mathbb{S}^d\cap H_1^+\cap H_2^+$ is a spherical convex subset of the $d$-dimensional unit sphere $\mathbb{S}^d$, which contains a great subsphere of dimension $d-2$ and is called a spherical wedge. Choose $n$ independent random points uniformly at random on $\mathbb{S}_{2,+}^d$ and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of $\log n$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on $\mathbb{S}_{2,+}^d$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.
