A well-known theorem of J. Heinonen and S. Rohde in 1993 states that if D ⊂ Rn is quasiconformally equivalently to an uniform domain, then the Gehring-Hayman inequality holds in D: quasihyperbolic geodesics in D minimizes the Euclidean length among all curves in D with the same end points, up to a universal dimension-dependent multiplicative constant. In this paper, we develop a new approach to strengthen the above result in the following three aspects: 1) obtain a dimension-free multiplicative constant in the Gehring-Hayman inequality; 2) relax the class of quasihyperbolic geodesics to more general quasigeodesics; 3) relax the quasiconformal equivalence to more general coarsely quasihyperbolic equivalence. As a byproduct of our general approach, we are able to prove that the above improved Gehring-Hayman inequality indeed holds in Banach spaces. This answers affirmatively an open problem raised by J. Heinonen and S. Rohde in 1993 and reformulated by J. Väisälä in 2005.

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