Metric definition of quasiconformality and exceptional sets

Abstract

We show that a homeomorphism of Euclidean space is quasiconformal if and only if at each point there exists a sequence of uncentered open sets with bounded eccentricity shrinking to that point whose images also have bounded eccentricity. This generalizes the metric definition of quasiconformality of Gehring that uses balls instead. We also study exceptional sets for this definition, in connection with sets that are negligible for extremal distances. We introduce the class of CNED sets, generalizing the classical notion of NED sets studied by Ahlfors--Beurling. A set $A$ is CNED if the conformal modulus of a curve family is not affected when one restricts to the subfamily intersecting the set $A$ at countably many points. We show as our main theorem that CNED sets are exceptional for the definition of quasiconformality.