Geometric probability on the sphere
This is a beautiful, expository paper containing many interesting results from geometric probability on the surface of a ball in three-dimensional space. The authors carefully develop the geometric machinery needed to present their arguments in an intuitive, elementary and elegant way. In Section 2, basics of spherical geometry are recalled including Lambert's cylindrical projection, convex polygons/sets on the sphere, polar sets and the duality theorem. The authors use these tools to study random convex polygons on the sphere in Section 3. In particular, they provide expectations for the perimeter and area of spherical triangles determined by three random points as well as probabilities for the shape of the convex hull of 4 and 5 random points. The authors remark on difficulties when increasing the number of points and formulate two problems for future research. In Section 4, Crofton's formula as well as Santaló's chord theorem are derived. This sets the stage for the study of random spherical caps in Section 5. For example, in Theorem 5.1 a lower bound is given for the probability that the union of \(N\) random spherical caps is connected and two more open problems are formulated. The authors also discuss the coverage problem, i.e., given \(N\) spherical caps of the same size, what is the probability that they cover the sphere? The paper is concluded in Section 6 with a glimpse into higher dimensions.
scientific article; zbMATH DE number 6714134
scientific article