Analytic capacity: discrete approach and curvature of measure

Certain discrete `computable' quantities are introduced, and their interconnections and relations with analytic capacity are found out. The concept of curvature of a measure is introduced, which emerges naturally in the computations of the -norm of the Cauchy transform of this measure. A lower bound on the analytic capacity, which uses the measure curvature and which has, to this extent, a geometric nature, is obtained.