This chapter concerns various geometric applications of Fourier series that either do not have higher dimensional analogues or serve as good illustrations for the methods used in the more complicated d-dimensional case. For a survey of the results discussed here see Groemer (1993c, chapter 2). Although most of these results are relatively old, some of the proofs have been modified to avoid smoothness assumptions quite often present (explicitly or implicitly) in the original literature.

A Proof of Hurwitz of the Isoperimetric Inequality

The aim of this section is to present a proof of the isoperimetric inequality (in E2) based on the ideas of the classical paper of Hurwitz (1901). It is remarkable that this proof can be arranged in such a way that no smoothness assumption and not even convexity are required.

We first discuss a few concepts and known results regarding curves in E2 that will be used here. A curve is defined as a continuous mapping of a closed interval [α, β] into E2 that is not constant on any subinterval of [α, β]. In this connection intervals are always assumed to have positive length. Any two such curves, say Γ1 and Γ2, are considered to be the same if Γ2 is obtained from Γ1 by an admissible change of parameter.