Integral geometry from Buffon to geometers of today
type: Item
instance of: scholarly article
zbMATH Open document ID: 1353.53004
author: Rémi Langevin
publication date: 2016-04-12T00:00:00Z
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 52A22
zbMATH DE Number: 6567429
zbMATH Keywords: Grassmann manifold
zbMATH Keywords: Cauchy-Crofton theorem
zbMATH Keywords: Gauss map
zbMATH Keywords: total curvature
zbMATH Keywords: hérisson
zbMATH Keywords: flag space
zbMATH Keywords: polyhedral surface
zbMATH Keywords: exchange theorem
zbMATH Keywords: knot
zbMATH Keywords: Clifford torus
zbMATH Keywords: genus
zbMATH Keywords: tight immersion
zbMATH Keywords: foliation
zbMATH Keywords: Reeb foliation
zbMATH Keywords: Chern-Lashof inequality
zbMATH Keywords: link
zbMATH Keywords: Dupin cyclide
zbMATH Keywords: Willmore conjecture
zbMATH Keywords: Quermassintegral
MaRDI profile type: Publication
title: Integral geometry from Buffon to geometers of today
published in: Cours Spécialisés (Paris)
review text: This book presents an exposition of the most important results in the historical development of integral geometry. The origins of this area of mathematics go back to 1777, when Buffon published his \textit{Essai d'arithmétique morale}. There he proves that when a needle is randomly thrown on the boards of a parquet, if the length of the needle is equal to the width of the boards, the probability to lie across two boards is \(\frac{2}{\pi}\). The study of this problem naturally leads to the notion of a measure on the set of all possible positions of the needle. In 1832, Cauchy noticed that the length of a convex curve is proportional to the average of the lengths of its orthogonal projection on all lines through a fixed point, thus establishing what is now called \textit{Cauchy's formula}. Again, a measure on a set of lines is considered. However, it was a serious setback for the developing integral geometry, when Bertrand constructed his paradoxes, with which the author closes the first, introductory chapter.NEWLINENEWLINEIn the second chapter, the author introduces a measure on the set of affine straight lines in the plane and proves the Cauchy-Crofton theorem and a so-called exchange theorem. The first one states that the length of a curve is half of the average number of its intersection points with affine lines; the second one claims that the total curvature of a curve equals the integral (over all line directions) of the number of critical points of projections of the curve on lines.NEWLINENEWLINEThe third chapter is devoted to planar convex sets, their support functions, and Minkowski sums. The concept of a so-called hérisson is applied, which is an envelope of a family of lines parametrized by the angle of their normal with respect to a fixed axis.NEWLINENEWLINEIn the fourth chapter, the author introduces an invariant measure on the Lie group of affine isometries of the plane, in order to derive a kinematic formula of Poincaré relating the lengths of two comp…
reviewed by: Eleutherius Symeonidis
description: scientific article; zbMATH DE number 6567429