Conformal dimension and boundaries of planar domains

Rectifiability of planes and Alberti representations

“…Proof. This is proven in Proposition 5.4 of [Kin15], in the case where C has a single element X (in which case the Gromov-Hausdorff limits of pointed rescalings of X are called "weak tangents" of X). However, an identical proof works under our assumption that all elements of C are metrically Ddoubling, since the same compactness argument can be run.…”

“…It remains to prove Lemma A.3. To do so, we will use the following Lemma, which is a minor modification of Proposition 5.4 of [Kin15].…”

“…However, an identical proof works under our assumption that all elements of C are metrically Ddoubling, since the same compactness argument can be run. (Note that the boundedness assumption in Proposition 5.4 of [Kin15] is not needed here, because we allow arbitrary scalings in the hypotheses of Proposition A.3.) Lemma A.5.…”