Geometric Applications of Fourier Series and Spherical Harmonics [Paperback]

A full exposition of the classical theory of spherical harmonics and their use in proving stability results.This self-contained, comprehensive treatise presents a careful introduction to the classical theory of spherical harmonics and shows how this theory can be used to prove geometric results such as geometric inequalities, uniqueness results for projections and intersection by hyperplanes or half-spaces, and stability. The analytic nature of the proofs is emphasized, since this makes them particularly useful in applications. Many of the results appear here in book form for the first time. This reference will be welcomed by both pure and applied mathematicians.This self-contained, comprehensive treatise presents a careful introduction to the classical theory of spherical harmonics and shows how this theory can be used to prove geometric results such as geometric inequalities, uniqueness results for projections and intersection by hyperplanes or half-spaces, and stability. The analytic nature of the proofs is emphasized, since this makes them particularly useful in applications. Many of the results appear here in book form for the first time. This reference will be welcomed by both pure and applied mathematicians.This is the first comprehensive exposition of the application of spherical harmonics to prove geometric results. The author presents all the necessary tools from classical theory of spherical harmonics with full proofs. Groemer uses these tools to prove geometric inequalities, uniqueness results for projections and intersection by planes or half-spaces, stability results, and characterizations of convex bodies of a particular type, such as rotors in convex polytopes. Results arising from these analytical techniques have proved useful in many applications, particularly those related to stereology. To make the treatment as self-contained as possible the book begins with background material in analysis and the geometry of convex sets.Preface; 1. Analyl³¶