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Selected Topics in Integral Geometry
I. M. Gelfand and S. G. Gindikin, Rutgers University, New Brunswick, NJ, and M. I. Graev, Institute of System Studies, RAS, Moscow, Russia
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Translations of Mathematical Monographs
2003; 170 pp; hardcover
Volume: 220
ISBN-10: 0-8218-2932-7
ISBN-13: 978-0-8218-2932-5
List Price: US$94
Member Price: US$75.20
Order Code: MMONO/220
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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

  • Radon transform
  • John transform
  • Integral geometry and harmonic analysis on the hyperbolic plane and in the hyperbolic space
  • Integral geometry and harmonic analysis on the group \({G=SL(2,\mathbb C)}\)
  • Integral geometry on quadrics
  • Bibliography
  • Index
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