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 andvarious capacities, and increases the torsional rigidity and mean lifetime of a Brownian particle.

Chapter 5 covers three classical topics in symmetrization, and includes historical remarks as well as the needed background in physics to guide the reader. The first result is that symmetrizing a fixed membrane into a disk of the same area decreases its principal frequency (the first eigenvalue of the Laplacian with Dirichlet boundary conditions), as conjectured by Rayleigh in 1877 and proved independently by Faber and Krahn. The second result is that symmetrization increases the torsional rigidity of a planar domain, as conjectured by St. Venant in 1856 and proved by Pólya.Lastly, a closed ballin three dimensionsal space is shown to have the smallest Newtonian capacity among all compact sets with the same volume. This conjecture was raised by Poincaré in 1887 and proved by Szegö. The proofs depend on the decrease of the Dirichlet integral under symmetric decreasing rearrangement of the function