ON THE ANALYTIC CAPACITY AND CURVATURE OF SOME CANTOR SETS WITH NON-σ-FINITE LENGTH

We show that if a Cantor set E as considered by Garnett in [G2] has positive Hausdorff h-measure for a non-decreasing function h satisfying integral(1)(0) r(-3) h(r)(2) dr < infinity, then the analytic capacity of E is positive. Our tool will be the Menger three-point curvature and Melnikov's identity relating it to the Cauchy kernel. We shall also prove some related more general results.