The semiadditivity of continuous analytic capacity and the inner boundary conjecture

Let α(E) be the continuous analytic capacity of a compact set E ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. In this paper we obtain a characterization of α in terms of curvature of measures with zero linear density, and we deduce that α is countably semiadditive. This result has important consequences for the theory of uniform rational approximation on compact sets. In particular, it implies the so-called inner boundary conjecture.