Geometric considerations

Abstract

The purpose of this chapter is to discuss various geometric problems which are informed by orthogonality and related considerations. We begin with Hurwitz’s proof of the isoperimetric inequality using Fourier series. We prove Wirtinger’s inequality, both by Fourier series and by compact operators. We continue with a theorem comparing areas of the images of the unit disk under complex analytic mappings. We again give two proofs, one using power series and one using Green’s (Stokes’) theorem. The maps $z \mapsto z^d$ from the circle to itself play a prominent part in our story. We naturally seek the higher dimensional versions of some of these results. It turns out, not surprisingly, that one can develop the ideas in many directions. We limit ourselves here to a small number of possible paths, focusing on the unit sphere in $\mathbf{C}^n$, and we travel only a small distance along each of them.