A lecture on integral geometry

Integral geometry originated from geometric probability. It studies random geometric objects in a probability space endowed with a measure that is invariant under a group of transformations. The geometric objects are points, lines, planes, solids, curves, surfaces, geodesics, etc. Works of Crofton, Poincaré, Sylvester, and others set up the early stage of integral geometry. Poincaré defined the basic concept of kinematic density by considering it as a group invariant. This led to the study of measures on geometric objects invariant under a group. Integral geometry has developed into an elegant mathematical discipline by the works of Blaschke, Weil, Chern, Santaló, and others. It is closely related to convex geometry and global differential geometry. It is fundamental for stochastic geometry, geometric tomography and stereology. Integral geometry has found applications in many subjects other than mathematics, such as imaging science, material science, biological science, medical science, and information science. Santaló’s book [6] in 1976 is a milestone of integral geometry and is still the most important book on the subject. Howard’s book [2] in 1993 studies integral formulas in Riemannian geometry. The book [4] of Klain and Rota in 1997 explains the connections of integral geometry and convex geometry. The books [3, 5, 9] deal with various geometric probabilities in the Euclidean space. The book [8] by Schneider and Weil in 2008 includes recent