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

Crofton formulae and geodesic distance in hyperbolic spaces