Integral Geometric Transformations

Mean value formulas with respect to invariant measures, as treated in the preceding two chapters, are a central topic of integral geometry. Another one is transformation formulas for integrals over various spaces of geometric objects. The need for such results in stochastic geometry can be demonstrated by simple examples. Consider, for instance, two independent, identically distributed random hyperplanes in R^d. Suppose the distribution is such that the intersection of the two hyperplanes is almost surely a (d − 2)-flat. What is the distribution of this random (d − 2)-flat? Or, consider k ≤ d independent, identically distributed random points in R^d, and suppose their distribution is such that they almost surely span a (k − 1)-flat. What is its distribution? In the cases where the original distributions are derived from invariant measures (by restriction, for example), the answers can be obtained from simple cases of the transformation formulas of this chapter. Generally, these transformation formulas relate integrations over tuples of flats, with respect to invariant measures, to integrations over other sets of flats (or other geometric objects) that are obtained by geometric operations, such as intersection or span. As an example, consider the integral of a function depending on d points. It may happen that the function depends only on the hyperplane spanned (almost everywhere) by the points. Then it may have a simplifying effect to integrate first over the d-tuples of points lying in a fixed hyperplane, and then over all hyperplanes. In principle, the required transformation formulas are just versions of the transformation rule for multiple integrals under differentiable mappings. However, since the mappings are defined by geometric operations, the Jacobians have geometric interpretations, and therefore direct geometric arguments are often simpler and more perspicuous than the use of special parametrizations.