In this text, integral geometry deals with Radon's problem of
representing a function on a manifold in terms of its integrals over
certain submanifolds--hence the term the Radon transform. Examples and
far-reaching generalizations lead to fundamental problems such as: (i)
injectivity, (ii) inversion formulas, (iii) support questions, (iv)
applications (e.g., to tomography, partial di erential equations and
group representations). For the case of the plane, the inversion theorem
and the support theorem have had major applications in medicine through
tomography and CAT scanning. While containing some recent research, the
book is aimed at beginning graduate students for classroom use or
self-study. A number of exercises point to further results with
documentation. From the reviews: "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 vibration,
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