Integral Geometry is a fascinating area, where numerous branches of mathematics
meet together. The contents of the book is concentrated around the duality and double
fibration, which is realized through the masterful treatment of a variety of examples.
The book is written by an expert, who has made fundamental contributions to the area.
-- Boris Rubin, Louisiana State University
The book under review is an excellent introduction to the group theoretical and
analytic aspects of the field by one of its pioneers. Before reviewing the book, we
will provide an overview of the field.
Integral geometry draws together analysis, geometry, and numerical mathematics. It has direct applications in PDEs, group representations, and the applied mathematical field of tomography. The fundamental problem in integral geometry is to determine properties of a function f in the plane or three-dimensional space or other manifolds from knowing the integrals of f over lines, planes, hyperplanes, spheres, or other submanifolds.
-- Fulton Gonzalez & Eric Todd Quinto, Tufts University
In view of this wisdom, the book under review is evidently offered as a means whereby
to prepare properly for work on a theme in geometric analysis going back to the work of
Funk and Radon nearly a century ago, namely that of “determining, respectively, a
symmetric function on the two-sphere... from its great circle integrals and an integrable
function on [the plane] from its straight line integrals.” This theme was later taken up by
Fritz John, who “found significant applications to differential equations.”
-- Michael Berg, Loyola Marymount University
Helgason Buch The Radon Trsnsform von 1980 kann man inzwischen getrost als einen
Klassiker bezeichnen. Hier kommt nun eine erweiterte Version, die sich im Gegensatz zum Klassiker
vor allem an fortgeschrittene Studierende und jene Mathematiker wendet, die nicht allzuviel
Vorwissen über Integralgeometrische Transformation mitbringen.
-- M. Reitzner, Osnabruck