The miracle of integral geometry is that it is often
possible to recover a function on a manifold just from the knowledge of its
integrals over certain submanifolds. The founding example is the Radon
transform, introduced at the beginning of the 20th century. Since then, many
other transforms were found, and the general theory was developed. Moreover, many
important practical applications were discovered, the best known, but by
no means the only one, being to medical tomography.
The present book is a general introduction to integral geometry, the first
from this point of view for almost four decades. The authors, all
leading experts in the field, represent one of the most influential schools in
integral geometry. The book presents in detail basic examples of integral
geometry problems, such as the Radon transform on the plane and in space,
the John transform, the Minkowski–Funk transform, integral geometry on the
hyperbolic plane and in the hyperbolic space, the horospherical transform and
its relation to representations of $SL(2,\mathbb C)$, integral
geometry on quadrics, etc. The study of these examples allows the authors to
explain important general topics of integral geometry, such as the
Cavalieri conditions, local and nonlocal inversion formulas, and
overdetermined problems. Many of the results in the book
were obtained by the authors in the course of their career-long work in
integral geometry.
Readership
Graduate students and research mathematicians interested in
integral geometry and applications.