Geometric TomographyCambridge University Press, 29 sept 1995 - 424 páginas 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. It overlaps with convex geometry, and employs many tools from that area including integral geometry. It also has connections to geometric probing in robotics and to stereology. The main text contains a rigorous treatment of the subject starting from basic concepts and moving up to the research frontier: seventy-two unsolved problems are stated. Each chapter ends with extensive notes, historical remarks, and some biographies. This comprehensive work will be invaluable to specialists in geometry and tomography; the opening chapters can also be read by advanced undergraduate students. |
Índice
Background material | 1 |
Parallel Xrays of planar convex bodies | 27 |
Parallel Xrays in n dimensions | 60 |
Projections and projection functions | 98 |
Projection bodies and volume inequalities | 128 |
Point Xrays | 167 |
Chord functions and equichordal problems | 203 |
Sections section functions and point Xrays | 239 |
Estimates from projection and section functions | 311 |
Appendixes | 347 |
B Inequalities | 366 |
Integral transforms | 376 |
References | 387 |
Notation | 408 |
| 416 | |
Intersection bodies and volume inequalities | 270 |
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Términos y frases comunes
affine affine transformation area measures body of constant Borel measure boundary bounded Busemann centered convex body centrally symmetric convex centroid Chapter chord compact convex set compact set component constant width containing the origin convex polygon Corollary corresponding defined Definition denote determined directed X-rays endpoint equation example Figure finite set follows formula geometry Hausdorff Hausdorff metric homothetic hyperplane i-chord functions implies integral interior intersection body isoperimetric inequality k-dimensional K₁ and K2 K₂ L₁ L1 and L2 Lemma line segment linear Lutwak measurable sets Minkowski Note obtain orthogonal p₁ and p2 planar convex bodies plane point X-rays polar body polytope Problem projection body proof of Theorem proved radial function Radon transform result section function set of directions shows Steiner symmetral subset subspace support function Suppose surface area translate triangle unique vector vertices Volčič x-axis zonoid zonotope