Integral geometry

Summary

Integral geometry studies measures on families of geometric objects, such as the lines or the planes of Euclidean space. The measures are chosen so that they do not change under rigid motions. Its main results express a geometric size, such as the length of a curve, as an average over such a family. For example, the length of a curve is an average of the number of times a random line meets it. The subject grew out of geometric probability, starting with Buffon's needle problem, and is closely tied to convex geometry and to stochastic geometry.