Synopses & Reviews
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.
Review
"The author's attention to detail and insistence on complete proofs make this book an excellent resource. In addition to the selected major results, each section closes with notes which provide the historical background and references to, and discussion of other realted results. This is indeed a comprehensive presentation of the subject matter with much to offer both to the beginner and the expert." P.R. Goodey, Mathematical Reviews
Synopsis
This book provides a comprehensive presentation of geometric results, primarily from the theory of convex sets, that have been proved by the use of Fourier series or spherical harmonics. Almost all these geometric results appear here in book form for the first time. An important feature of the book is that all the necessary tools from classical theory of spherical harmonics are developed as concretely as possible, with full proofs. These tools are used to prove geometric inequalities, stability results, uniqueness results for projections and intersections by hyperplanes or half-spaces, and characterizations of rotors in convex polytopes. Again, full proofs are given. To make the treatment as self-contained as possible the book begins with background material in analysis and the geometry of convex sets. This treatise will be welcomed both as an introduction to the subject and as a reference book for pure and applied mathematicians.
Synopsis
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.
Description
Includes bibliographical references (p. [311]-318) and indexes.
Table of Contents
Preface; 1. Analytic preparations; 2. Geometric preparations; 3. Fourier series and spherical harmonics; 4. Geometric applications of Fourier series; 5. Geometric applications of spherical harmonics; References; List of symbols; Author index; Subject index.