Journal of Lie Theory Volume 8 (1998) 441 ◯^{C}1998 Heldermann Verlag

Addendum to: Crofton formulae and geodesic distance in hyperbolic spaces

Guyan Robertson^{∗}

Communicated by A. Valette

In Remark 2.6 of [1] we asked whether one could find a direct proof of the fact that the distance function on quaternionic hyperbolic space is a kernel of negative type. This is indeed possible, by considering the following 24 points. For σ ∈ {+1, −1} and ϵ ∈ {±i, ±j, ±k}, let x_{ϵ}^{σ} = (3, 2σ + 2ϵ, 0) and y_{ϵ}^{σ} = (3, 0, 2σ + 2ϵ). Direct calculation shows that

∑ d(x_{ϵ}^{σ}, x_{δ}^{ρ}) + ∑ d(y_{ϵ}^{σ}, y_{δ}^{ρ}) − ∑ d(x_{ϵ}^{σ}, y_{δ}^{ρ}) = 417.03 − 415.77 > 0.

This shows that the condition for negative type fails with the number +1 being associated to the x_{ϵ}^{σ} and −1 to the y_{ϵ}^{σ}.

We take the opportunity to point out that there is a misprint in Remark 2.6: the word “Hilbert” should be replaced by the word “hyperbolic.”

References

[1] Robertson, G., Crofton formulae and geodesic distance in hyperbolic spaces, Journal of Lie Theory 8 (1998), 163–172.

Department of Mathematics University of Newcastle NSW 2308 AUSTRALIA guyan@maths.newcastle.edu.au

Received December 19, 1997

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag

∗ This research was supported by the Australian Research Council.