**
Journal of Lie Theory**

8(1), 163-172 (1998)

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Crofton formulae and geodesic distance in hyperbolic spaces

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A. G. Robertson

Guyan Robertson

Department of Mathematics

University of Newcastle

NSW 2308

AUSTRALIA

guyan@maths.newcastle.edu.au

**Abstract:** 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.

**Classification (MSC91):** 22D10

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