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.