Weighted analogues of capacity,transfinite diameter,and Chebyshev constant

For an arbitrary closed subsetE of the complex plane, the notions of logarithmic capacity, transfinite diameter, and Chebyshev constant ofE with respect to an admissible weightw onE are introduced. For thew-modified capacity, an electrostatics problem for logarithmic potentials in the presence of an external field is analyzed. This leads to an extremal measure whose support is the “smallest” compact set where the sup norm of weighted polynomials “live.” The introduction of a weightw has the advantage that the classical quantities mentioned in the title can be considered for unbounded setsE. Some of the theorems presented are generalizations of the authors' previous results for the case whenE?R.