Integral geometry

Definition

Let $G$ be a group acting on a space $X$. Integral geometry is the study of $G$-invariant measures on spaces of geometric objects in $X$. It also studies the integrals of geometric quantities against these measures. In the classical case, $X$ is the Euclidean space $\R^n$ and $G$ is the Euclidean group of rigid motions. The objects are then the affine subspaces of a fixed dimension.

The basic example is the measure on lines in $\R^2$. For $\theta \in [0, \pi)$ and $p \in \R$, let $L(\theta, p) = \set{x \in \R^2 : x_1 \cos \theta + x_2 \sin \theta = p}$; every line is $L(\theta, p)$ for exactly one pair $(\theta, p)$. Transport the measure $dp \, d\theta$ from $[0, \pi) \times \R$ to the set of lines by this parametrization. The result is invariant under all rigid motions of $\R^2$, and up to a constant factor it is the only such measure.