Integral geometry and geometric probability

Book · Thu Jan 01 04:00:00 EST 1976

OSTI ID:7227539

Santalo, L A

This monograph is the first in a projected series on Probability Theory. Part I of the monograph is concerned with integral geometry on the euclidean plane, treated in an elementary way. The theory of geometric probability, sets of strips as an immediate generalization of sets of lines, and the kinematic measure in the plane are treated, along with some discrete subgroups of the motions group and their interpretation from the integral geometric viewpoint. Part II presents an account of the necessary elements of the theory of Lie groups and homogeneous spaces in order to obtain the invariant measures in these spaces and their properties. The general theory is exemplified by the groups of affine transformations and the group of motions in euclidean space. Part III is concerned with integral geometry in euclidean n-dimensional space. It contains a resume of the main results on convex sets in such a space. The measure of linear spaces which intersect a compact manifold embedded in euclidean space and the so-called kinematic fundamental formula are discussed. The general theory is applied in detail to three-dimensional euclidean space. Part IV deals with integral geometry in spaces of constant curvature (noneuclidean integral geometry), in particular integral geometry on the sphere, and some new trends in integral geometry (integral geometry and foliated spaces, integral geometry in complex spaces, symplectic integral geometry, and integral geometry in the sense of Gelfand and Helgason). (RWR)

• OSTI ID: 7227539
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
EUCLIDEAN SPACE
GEOMETRY
INTEGRALS
LIE GROUPS
MATHEMATICAL SPACE
MATHEMATICS
PROBABILITY
RIEMANN SPACE
SPACE
SYMMETRY GROUPS