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The Gehring-Hayman identity for the diameter of quasihyperbolic geodesics in convex domain. (Chinese. English summary) Zbl 1299.30119
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 Möbius 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.
##### MSC:
 30F45 Conformal metrics (hyperbolic, Poincaré, distance functions) 30C20 Conformal mappings of special domains 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations
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