WEIGHTED ANALOGS OF CAPACITY, TRANSFINITE DIAMETER, AND CHEBYSHEV CONSTANT

For an arbitrary closed subset E of the complex plane, the notions of logarithmic capacity, transfinite diameter, and Chebyshev constant of E with respect to an admissible weight w on E are introduced. For the w-modified capacity, an electrostatics problem for logarithmic potentials in the presence of an external field is analyzed. This leads to an external measure whose support is the "smallest" compact set where the sup norm of weighted polynomials "live." The introduction of a weight w has the advantage that the classical quantities mentioned in the title can be considered for unbounded sets E. Some of the theorems presented are generalizations of the authors' previous results for the case when E subset-of R.