Geometric probability on the sphere
type: Item
instance of: scholarly article
title: Geometric probability on the sphere
zbMATH Open document ID: 1401.60019
author: H. Maehara
author: Horst Martini
published in: Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)
publication date: 2017-05-11T00:00:00Z
review text: 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.
reviewed by: Florian Pausinger
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 52A22
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 60-02
zbMATH DE Number: 6714134
zbMATH Keywords: coverage problem
zbMATH Keywords: Crofton's formula
zbMATH Keywords: random great circle
zbMATH Keywords: random spherical polygon
zbMATH Keywords: Santaló's chord theorem
zbMATH Keywords: Sylvester's problem
MaRDI profile type: Publication
full work available at URL: s13291-017-0158-5
OpenAlex ID: W2582613430
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DOI: 10.1365/S13291-017-0158-5
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