We complete the proof of a conjecture of Vitushkin that says that if is a compact set in the complex plane with finite 1-dimensional Hausdorff measure, then has vanishing analytic capacity (i.e., all bounded analytic functions on the complement of are constant) if and only if is purely unrectifiable (i.e., the intersection of with any curve of finite length has zero 1-dimensional Hausdorff measure). As in a previous paper with P. Mattila, the proof relies on a rectifiability criterion using Menger curvature and an extension of a construction of M. Christ. The main new part is a generalization of the -Theorem to some spaces that are not necessarily of homogeneous type.
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Guy David, Unrectictifiable 1-sets have vanishing analytic capacity. Rev. Mat. Iberoam. 14 (1998), no. 2, pp. 369–479DOI 10.4171/RMI/242