Geometric tomography / Richard J. Gardner.

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

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

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.