Capacity, virtual mass, and generalized symmetrization

Abstract

A body of revolution B can be symmetrized with respect to its axis of symmetry in a number of ways. One of these is the Schwarz symmetrization, which preserves the volume of B. Another is the Steiner symmetrization of the meridian section of B, which preserves the area of this section but in general decreases the volume. The influence of the Schwarz symmetrization on the capacity has been investigated by G. Polya and G. Szeg*, [l]. More recently P. R. Garabedian and D. C. Spencer [2] discussed the same question for the virtual mass of bodies of revolution. In the present paper we shall study by a different and simpler method the behavior of the capacity and virtual mass under a more general type of symmetrization, which includes the Schwarz and Steiner symmetrizations as particular cases.