**Translations of Mathematical Monographs**

2003;
170 pp;
Hardcover

MSC: Primary 53;
Secondary 42; 43; 44; 46; 60; 65; 92

Print ISBN: 978-0-8218-2932-5

Product Code: MMONO/220

List Price: $99.00

AMS Member Price: $79.20

MAA Member Price: $89.10

**Electronic ISBN: 978-1-4704-4644-4
Product Code: MMONO/220.E**

List Price: $99.00

AMS Member Price: $79.20

MAA Member Price: $89.10

# Selected Topics in Integral Geometry

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*I. M. Gelfand; S. G. Gindikin; M. I. Graev*

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.

This 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 in integral
geometry. Many of the results in the book were obtained by the authors in the
course of their career-long work in integral geometry.

This book is suitable for graduate students and researchers working in
integral geometry and its applications.

#### Readership

Graduate students and research mathematicians interested in integral geometry and applications.

#### Table of Contents

# Table of Contents

## Selected Topics in Integral Geometry

- Cover Cover11
- Title page iii4
- Contents v6
- Preface to the English edition xi12
- Preface xiii14
- Radon transform 118
- John transform 4360
- Integral geometry and harmonic analysis on the hyperbolic plane and in the hyperbolic space 7996
- Integral geometry and harmonic analysis on the group 𝐺=𝑆𝐿(2,ℂ) 111128
- Integral geometry on quadrics 145162
- Bibliography 165182
- Index 167184
- Back Cover Back Cover1190