In 1901 Adolf Hurwitz published a short note showing that Fourier series can be used to prove the isoperimetric inequality for domains in the Euclidean plane, and in a subsequent article he showed how spherical harmonics can be utilized to prove an analogous inequality for three-dimensional convex bodies. A few years later Hermann Minkowski used spherical harmonics to prove an interesting characterization of (three-dimensional) convex bodies of constant width. The work of Hurwitz and Minkowski has convincingly shown that a study of this interplay of analysis and geometry, in particular of Fourier series and spherical harmonics on the one hand, and the theory of convex bodies on the other hand, can lead to interesting geometric results. Since then many articles have appeared that explored the possibilities of such methods.

The aim of the present book is to provide a fairly comprehensive exposition of geometric results, more specifically, of results in the theory of convex sets, that have been proved by the use of Fourier series or spherical harmonics. Almost all theorems that are stated are also proved. Furthermore, to make the book more self-contained, all results from the theory of spherical harmonics that are used are also proved. Thus the only prerequisite for reading this book is some familiarity with the basic facts of the theory of (finite dimensional) convex sets and the theory of functions of real variables.

In this chapter we discuss some of the basic concepts and facts regarding the geometry of convex sets. Additional definitions and notations that are of a more limited scope will be introduced when needed. In most cases no proofs are given since these are readily available in the standard textbook literature dealing with this subject area. In particular we mention the books of Bonnesen and Fenchel (1934), Hadwiger (1957), Eggleston (1958), Valentine (1964), Leichtweiss (1980), Schneider (1993b), and Webster (1995). In fact, a large portion of this material, at least in the three-dimensional case, can already be found in the original work of Minkowski (1903, 1911). If a particular result is of importance for our objectives and if it is not textbook material we include a proof.

Basic Features of Convex Sets

As before, Ed denotes the Euclidean space of dimension d (d ≥ 2) whose points are of the form x = (x1, …, xd) and whose origin is o = (0, …, 0). The boundary and interior of a subset X of Ed will be denoted by ∂X and int X, respectively. A nonempty compact convex subset of Ed will be called a convex body or, more specifically, a convex body in Ed, and the class of all convex bodies in Ed will be denoted by Κd. If it is necessary to indicate that a convex body in Ed has interior points it will be referred to as a d-dimensional convex body.

In this chapter we develop the theory of spherical harmonics to the extent necessary for our geometric applications. Occasionally, if it seems helpful for the understanding of the subject area, some topics will be developed in more detail or with a more general point of view than absolutely necessary for applications. As in the previous chapters it is always assumed that d ≥ 2. In some formulas presented in this chapter, particularly in Sections 3, 4, and 5, there arise products that are, strictly speaking, meaningless for certain values of the integers appearing in them; for example (d + 1)(d + 2) … (d + n − 1) if n = 1. Unless something else is explicitly stated, in all such situations the value of the product is defined to be 1.

From Fourier Series to Spherical Harmonics

We first list here a few basic facts from the theory of Fourier series. More precisely, we should say trigonometric or classical Fourier series since the term “Fourier series” has already been used in Section 1.1 in a more general setting. But it will always be clear from the context which kind of Fourier series is meant. It is not necessary to include here any proofs, since all the listed results are either well-known facts of basic real analysis or will be proved later in the more general context of spherical harmonics.