Integral Geometry and Geometric Probability

ISBN-13: 9780521523448 / Angielski / Miękka / 2004 / 428 str.

Luis Santalo; Mark Kac

Opis

Integral 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.

Spis treści

Part I. Integral Geometry in the Plane: 1. Convex sets in the plane; 2. Sets of points and Poisson processes in the plane; 3. Sets of lines in the plane; 4. Pairs of points and pairs of lines; 5. Sets of strips in the plane; 6. The group of motions in the plane: kinematic density; 7. Fundamental formulas of Poincaré and Blaschke; 8. Lattices of figures; Part II. General Integral Geometry: 9. Differential forms and Lie groups; 10. Density and measure in homogenous spaces; 11. The affine groups; 12. The group of motions in En; Part III. Integral Geometry in En: 13. Convex sets in En; 14. Linear subspaces, convex sets and compact manifolds; 15. The kinematic density in En; 16. Geometric and statistical applications: stereology; Part IV. Integral Geometry in Spaces of Constant Curvature: 17. Noneuclidean integral geometry; 18. Crofton's formulas and the kinematic fundamental formula in noneuclidean spaces; 19. Integral geometry and foliated spaces: trends in integral geometry.