Crofton formulae and geodesic distance in hyperbolic spaces

The geodesic distance between points in real hyperbolic space is a hypermetric, and hence is a kernel negative type. The proof given here uses an integral formula for geodesic distance, in terms of a measure on the space of hyperplanes. An analogous integral formula, involving the space of horospheres, is given for complex hyperbolic space.By contrast geodesic distance in a projective space is not of negative type.