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Given a self-similar set K in W we prove that the strong open set condition and the open set condition are both equivalent to Ha(K) > 0, where a is the similarity dimension of K and H" denotes the Hausdorff measure of this dimension. As an application we show for the case a = s that K possesses inner points iff it is not a Lebesgue null set.doi:10.2307/2160849 fatcat:er73roj4azb2nchtgbcrymzo6m