CHAPTER 5.1 - Integral Geometry

Abstract

This chapter discusses integral geometry. Integral geometry is concerned with the study, computation, and application of invariant measures on sets of geometric objects. It has its roots in some questions on geometric probabilities. Integral geometry is closely connected to the geometry of convex bodies. The chapter describes some notation, concepts, and results concerning the main spaces occurring in the integral geometry of Euclidean spaces. The most familiar type of integral-geometric formula is the intersection of a fixed and a moving geometric object. For example, the principal kinematic formula for convex bodies provides an explicit expression for the measure of all positions of a moving convex body, in which it meets a fixed convex body K. Crofton's intersection formula does the same for the invariant measure of the set of all k-dimensional flats meeting a convex body. The functionals of convex bodies appearing in the results, the quermassintegrals or intrinsic volumes, can replace the characteristic functions in the integrations with respect to invariant measures. The resulting formulae can be further generalized, as they are valid in local versions—namely, for curvature measures.