Integral Geometry and Geometric ProbabilityIntegral geometry originated with problems on geometrical probability and convex bodies. Its later developments have proved to be useful in several fields ranging from pure mathematics (measure theory, continuous groups) to technical and applied disciplines (pattern recognition, stereology). The book is a systematic exposition of the theory and a compilation of the main results in the field. The volume can be used to complement courses on differential geometry, Lie groups, or probability or differential geometry. It is ideal both as a reference and for those wishing to enter the field. |
Contents
INTEGRAL GEOMETRY IN THE PLANE | 1 |
Part II GENERAL INTEGRAL GEOMETRY | 143 |
Part III INTEGRAL GEOMETRY IN En | 215 |
Part IV INTEGRAL GEOMETRY IN SPACES OF CONSTANT CURVATURE | 299 |
Bibliography and References | 363 |
| 395 | |
| 399 | |
Common terms and phrases
1-forms a₁ angle area F Assume boundary bounded convex set C₁ Chapter chord circle consider constant contained convex body convex set coordinates D₁ defined differentiable manifold differential form dimension distance dK₁ domain dP₁ dx₁ e₁ equal euclidean euclidean space exterior product F₁ fixed function fundamental regions geodesics geometric probability given Hence homogeneous space hyperplanes integral formula integral geometry integral of mean intersection points invariant density isoperimetric inequality K₁ K₂ kinematic density L. A. Santalo L₁ L₂ lattice left-invariant line G linear Math matrix Maurer-Cartan Maurer-Cartan forms mean curvature mean number mean value measure n-dimensional n.e. space number of intersection orthogonal P₁ parallel particles perimeter plane problem r-planes radius random lines Section subgroup t₁ t₂ tangent theorem triangle unimodular unit sphere V₁ vector volume element w₁ πα