Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross-sections by planes. It is a geometric relative of computerized tomography, which reconstructs an image from X-rays of a human patient. The subject overlaps with convex geometry and employs many tools from that area, including some formulas from integral geometry. It also has connections to discrete tomography, geometric probing in robotics and to stereology. This comprehensive study provides a rigorous treatment of the subject. Although primarily meant for researchers and graduate students in geometry and tomography, brief introductions, suitable for advanced undergraduates, are provided to the basic concepts. More than 70 illustrations are used to clarify the text. The book also presents 66 unsolved problems. Each chapter ends with extensive notes, historical remarks, and some biographies. This edition includes numerous updates and improvements, with some 300 new references bringing the total to over 800.
By:
Richard J. Gardner (Western Washington University)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Edition: 2nd Revised edition
Volume: No. 58
Dimensions:
Height: 234mm,
Width: 156mm,
Spine: 29mm
Weight: 790g
ISBN: 9780521684934
ISBN 10: 0521684935
Series: Encyclopedia of Mathematics and its Applications
Pages: 514
Publication Date: 11 September 2006
Audience:
College/higher education
,
Professional and scholarly
,
Primary
,
Undergraduate
Format: Paperback
Publisher's Status: Active
Preface to the second edition; Preface; 0. Background material; 1. Parallel X-rays of planar convex bodies; 2. Parallel X-rays in n dimensions; 3. Projections and projection functions; 4. Projection bodies and volume inequalities; 5. Point X-rays; 6. Chord functions and equichordal problems; 7. Sections, section functions, and point X-rays; 8. Intersection bodies and volume inequalities; 9. Estimates from projection and section functions; Appendix A. Mixed volumes and dual mixed volumes; Appendix B. Inequalities; Appendix C. Integral transforms; References; Notation; Author index; Subject index.
Richard Gardner has been Professor of Mathematics at Western Washington University since 1991. He is the author of 70 papers and founded geometric tomography as a subject in its own right with the publication of the first edition of this book in 1995.
Reviews for Geometric Tomography
praise for the first edition ... 'The work is written in a very clear style, is well organized, and is amply provided with helpful illustrations and diagrams.' Wm. J. Firey, SIAM Review 'This is an extremely useful monograph, collecting together all the currently available material on geometric tomography ... The book is well organized and easy to read, with many illustrations, and copious notes to each chapter ... I recommend the book highly as a valuable introduction and continuing reference.' Peter McMullen, Bulletin of the London Mathematical Society '... A valuable reference source. Libraries should have this book.' A. G. Ramm, Inverse Problems Newsletter 'This well-written book describes the recent theory ... it includes copious notes and a comprehensive bibliography.' Mathematika 'The book is self contained, readable and can be warmly recommended not only for specialists.' European Mathematical Society Newsletter '... this book is a must for all those who are or plan to work on geometric tomography ...' Acta Scientiarum Mathematicarum
