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Geometric applications of Fourier series and spherical harmonics. (English) Zbl 0877.52002
Encyclopedia of Mathematics and Its Applications. 61. Cambridge: Cambridge Univ. Press. xi, 329 p. (1996).
Early this century, Hurwitz and Minkowski each published papers in which they applied Fourier series and spherical harmonics to geometric questions, related to isoperimetric inequalities and convex bodies. Ever since, these analytic tools have proved useful in treating special types of problems in the geometry of convex bodies in Euclidean spaces. The applicability of harmonic analysis on the sphere to such geometric problems rests, roughly, on two principles. Certain kinds of questions lead to linear integral or other functional equations on the sphere \(S^{d-1}\), which are rotation invariant and hence can be solved in general if they can be solved for the elements of the irreducible invariant subspaces of \(L_2(S^{d-1})\), that is, the spherical harmonics of a given order. A second device is the Parseval relation for the system of spherical harmonics, which can be employed to yield quadratic inequalities for geometric functionals, which can then be combined with other geometric inequalities.
This well-written book gives the first coherent account of the applications of harmonic analysis on \(S^{d-1}\) to the geometry of convex bodies (and, occasionally, star bodies). The first chapter collects analytic preliminaries, like orthogonal function systems, the Laplace operator on spheres, and spherical integration. Basic concepts from the geometry of convex bodies make up Chapter two. A few special results needed later are already included here, for example, inequalities connecting the Hausdorff and the \(L_2\)-distance between convex bodies. Chapter three gives an elementary and particularly neat introduction to spherical harmonics and the corresponding series expansions, including a thorough treatment of Legendre polynomials and associated Legendre functions. This introduction should also be of use for readers not aiming at the geometric applications. Some of the later applications are prepared by the proof of the Funk-Hecke theorem, its application to some linear integral transformations for functions on \(S^{d-1}\), and some estimates to be used for stability results.
The geometric applications in Chapter four are restricted to the planar case, where Fourier series suffice. Among the applications are the isoperimetric inequality for rectifiable Jordan curves, improvements and stability versions of the isoperimetric and mixed area inequalities for convex domains, Bonnesen-type inequalities, classes of convex bodies with special properties of circumscribed equiangular \(n\)-gons, in particular rotors, and more.
Chapter five, on geometric applications of spherical harmonics in \(d\)-space, is the principal part of the book. Here we first find various improvements, in the form of stability results, of inequalities between mixed volumes and quermassintegrals, especially of the isoperimetric inequality. Another section is devoted to estimates of the deviation of two convex bodies in terms of the deviation of their projection bodies of orders 1 or \(d-1\); also other problems related to projections are considered here. Further topics are uniqueness and stability problems for star bodies connected with intersections by linear subspaces or halfspaces. Another section is devoted to rotors in polytopes, and a final one collects miscellanea, like characterizations of ellipsoids, variations on Cauchy’s surface area formula, cylindrical mean values, and a characterization of the Steiner point.
The proofs, among which are some longer ones, are detailed and clear. References and remarks to each section contain much additional information. The book gives a good impression of the interesting interplay between analysis and geometry in a classical part of the theory of convex bodies. Recent work by several authors, employing also harmonic analysis on Grassmannians \(SO_d/SO_k\) or the rotation group \(SO_d\) itself, which is beyond the scope of the present book, shows that applications of harmonic analysis to convex geometry are still vivid. The present book gives an excellent introduction to this subject and a useful survey.

52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces