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The Gehring–Hayman inequality for quasihyperbolic geodesics

Published online by Cambridge University Press:  24 October 2008

Juha Heinonen  and
Steffen Rohde
Juha Heinonen
Affiliation:
University of Michigan, Department of Mathematics, Ann Arbor, MI 48109, USA
Steffen Rohde
Affiliation:
University of Michigan, Department of Mathematics, Ann Arbor, MI 48109, USA
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Extract

The quasihyperbolic metric in a proper subdomain D of Rn is defined by

where the infimum is taken over all rectifiable arcs γ in D joining x and y. There always exists an arc, called a quasihyperbolic geodesic in D, for which the infimum above is attained. We refer to [3], [4], [16], and [17] for the motivation and basic properties of the quasihyperbolic metric.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 3 , November 1993 , pp. 393 - 405
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

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