Kingman’s classical formula [1] from integral geometry establishes a relation
between the moments of the mean distance between two random points chosen
independently and uniformly inside a convex body in Rn, and the moments of the
length of its random section. We obtain an analogue of this formula in the case
when one of the points is chosen at random on the boundary of the body. As a
corollary, a relation between the distribution functions of the considered quantities
is derived. An analogous result for two points inside the body was obtained by
Moseeva [2].
The proof is based on the introduction of a new “homothetic” coordinate system
adapted to the geometry of K, combining ideas of stochastic geometry and the
theory of convex bodies.