On the subinvariance of uniform domains in Banach spaces

Suppose that EE and E′E′ denote real Banach spaces with dimension at least 22, that D⊂ED⊂E and D′⊂E′D′⊂E′ are domains, and that f:D→D′f:D→D′ is a homeomorphism. In this paper, we prove the following subinvariance property for the class of uniform domains: suppose that ff is a freely quasiconformal mapping and that D′D′ is uniform. Then the image f(D1)f(D1) of every uniform subdomain D1D1 in DD under ff is still uniform. This result answers an open problem of Väisälä in the affirmative.