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.

We review here some of the analytic concepts and facts that will be used in later chapters. Most of this material forms part of the standard textbook literature on real analysis or functional analysis and it is not necessary to repeat here the pertinent proofs. However, a few facts of a more special character and not generally known will be formulated as lemmas and proved.

Throughout this book we let Ed denote the Euclidean d-dimensional space. If x is a point of Ed the coordinates of x will be denoted by xi; hence, x = (x1, …, xd). The letter o denotes the origin (0, …, 0) of Ed. If u, v ∈ Ed we let u · v denote the inner product, and |u| the Euclidean norm. Of course, for points in E1, that is, for real numbers, | · | is the ordinary absolute value. The Lebesgue measure of a subset S of Ed will usually be called the volume of S and denoted by v(S). We write Bd(p, r) for the closed ball in Ed of radius r centered at p, and Bd = Bd(o, 1) for the closed unit ball in Ed centered at o. Furthermore, we let Sd–1 denote the boundary of Bd, that is, the unit sphere in Ed. The spherical Lebesgue measure on Sd–1 will be denoted by σ, the volume of Bd by κd, and the surface area of Bd by σd.