Geometric Tomography

Gardner, Richard J.
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.

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.

Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on 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 will be invaluable to specialists in geometry and tomography; opening chapters can also be read by advanced undergraduate students.

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; A. Mixed volumes and dual mixed volumes; B. Inequalities; C. Integral transforms.