Recently I was led to some considerations in geometric probability, a field pretty far from any specialization of mine. (Context: I was working with a collaborator on a question about mean escape time of a photon from a region in 3-space.)
From what I've gleaned from papers such as this one by A.M. Kellerer, there exist very nice and surprising results (due to Cauchy, Crofton, Czuber, and Hostinsky) that give the first and fourth moments for random chord length when a convex solid in Euclidean 3-space is intersected by a line passing through a uniformly distributed point and in a uniformly distributed direction. These results give the first and fourth moments in terms of simple ratios involving the volume and surface area of the convex solid. More specifically, the first moment is given by $\frac{4V}{S}$ and the fourth moment by $\frac{12V^2}{\pi S}$.
My question: Surely something deep is going on here, but what? Why are the first and fourth moments so easy to express in terms of simple geometric features of the convex solid, but not, say, the second moment?
