Illinois Journal of Mathematics

Hyperbolic space has strong negative type

Russell Lyons

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It is known that hyperbolic spaces have strict negative type, a condition on the distances of any finite subset of points. We show that they have strong negative type, a condition on every probability distribution of points (with integrable distance to a fixed point). This implies that the function of expected distances to points determines the probability measure uniquely. It also implies that the distance covariance test for stochastic independence, introduced by Székely, Rizzo and Bakirov, is consistent against all alternatives in hyperbolic spaces. We prove this by showing an analogue of the Cramér–Wold device.

Article information

Illinois J. Math., Volume 58, Number 4 (2014), 1009-1013.

Received: 4 October 2014
Revised: 2 March 2015
First available in Project Euclid: 6 November 2015

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Zentralblatt MATH identifier

Primary: 51K99: None of the above, but in this section 51M10: Hyperbolic and elliptic geometries (general) and generalizations
Secondary: 30L05: Geometric embeddings of metric spaces 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]


Lyons, Russell. Hyperbolic space has strong negative type. Illinois J. Math. 58 (2014), no. 4, 1009--1013. doi:10.1215/ijm/1446819297.

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