an:06264066
Zbl 1299.30119
Qian, Weimao; Zhang, Yichi
The Gehring-Hayman identity for the diameter of quasihyperbolic geodesics in convex domain
ZH
Pure Appl. Math. 29, No. 3, 241-245, 274 (2013).
00330131
2013
j
30F45 30C20 30C65
convex domain; quasihyperbolic length; quasihyperbolic distance; quasihyperbolic geodesics; Gehring-Hayman inequality
Summary: This paper generalizes the Gehring-Hayman inequality for the diameter of the hyperbolic geodesics in the plane Jordan domain to the quasihyperbolic geodesics in the convex domain of \(n\)-dimensional space. Making use of the M??bius transformation and the quasihyperbolic metric, we prove that the diameter of the quasihyperbolic geodesics with the endpoints \(x\) and \(y\) in the convex domain of \(n\)-dimensional space is equal to the Euclidean distance between \(x\) and \(y\). The obtained result is a generalization and improvement of some known results.