Quasispheres and metric doubling measures

Applying the Bonk-Kleiner characterization of Ahlfors 2-regular quasispheres, we show that a metric two-sphere \(X\) is a quasisphere if and only if \(X\) is linearly locally connected and carries a weak metric doubling measure, i.e., a measure that deforms the metric on \(X\) without much shrinking.