Steiner Symmetrization

Chapter 6 discusses Steiner symmetrization. Basic properties of symmetric decreasing rearrangement and polarization that were developed in Chapter 1 are adapted to Steiner symmetrization, to show that it decreases the modulus of continuity and acts contractively in L-Infinity. The effect of Steiner symmetrization on various Dirichlet integrals is studied. It is shown that Steiner symmetrization decreases perimeter and Minkowski content, but in general it is not known whether it decreases the (n-1)-dimensional Hausdorff measure. Steiner symmetrization also decreases the principal frequency and various capacities, and increases the torsional rigidity and mean lifetime of a Brownian particle.