Integral geometry

In mathematics, the term integral geometry is used in two ways, which, although related, imply different views of the content of the subject.

Cases
The more traditional usage is that of Santalo and Blaschke. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations.

There is a sample space of lines, one on which the affine group of the plane acts. A probability measure is sought on this space, invariant under the symmetry group. If, as in this case, we can find a unique such invariant measure, that solves the problem of formulating accurately what 'random line' means; and expectations become integrals with respect to that measure. (Note for example that the phrase 'random chord of a circle' can be used to construct some paradoxes.)

We can therefore say that integral geometry in the sense of Santalo, is the application of probability theory (as axiomatized by Kolmogorov) in the context of the Erlangen programme of Klein. The content of the theory is effectively that of invariant (smooth) measures on (preferably compact) homogeneous spaces of Lie groups; and the evaluation of integrals of differential forms arising.

A very celebrated case is the problem of Buffon's needle: drop a needle on a floor made of planks and calculate the probability the needle lies across a crack. Generalising, this theory is applied to various stochastic processes concerned with geometric and incidence questions.

One of the most interesting theorems in this form of integral geometry is Hadwiger's theorem.

The more recent meaning of integral geometry is that of Israel Gelfand. It deals more specifically with integral transforms, modelled on the Radon transform. Here the underlying geometrical incidence relation (points lying on lines, in Crofton's case) is seen in a freer light, as the site for an integral transform composed as pullback onto the incidence graph and then push forward.