It is known that the inverse of a quasiconformal homeomorphism of domains in R n is also quasiconformal. This paper focuses on the inverse problem for free quasiconformality in Banach spaces. We first show that the inverse of a fully semisolid homeomorphism is fully semisolid under an additional coarsely Lipschitz condition in the quasihyperbolic metric. This gives several partial answers to two open problems posed by V & auml;is & auml;l & auml;. Next, we prove that the inverse of a locally quasisymmetric homeomorphism is also locally quasisymmetric. As applications, we obtain new characteriz...
