On Two Problems in the Theory of Unique Determination of Domains

The estimates of the absolute and relative conformal moduli of the boundary capacitor with two parallel segments as its plates are given in the first part of this paper. The second one is devoted to a proof of the theorem which states that convexity of a plane domain $U$ with a smooth boundary doesn't lead, generally speaking, to a unique determination of $U$ in the class of all plane domains with smooth boundaries by the local isometry condition of boundaries in their relative metrics.