Geometric tomography

Title:

Personal Author:

Gardner, Richard J.

Series:

Encyclopedia of mathematics and its applications ; 58

Edition:

2nd ed.

Publication Information:

Cambridge, UK : Cambridge University Press, 2006

ISBN:

9780521684934

9780521866804

Subject Term:

Geometric tomography

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010152047	QA639.5 G37 2006	Open Access Book	Book	Searching... Unknown

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.

Preface to the second edition	p. xv
Preface	p. xvii
0 Background material	p. 1
0.1 Basic concepts and terminology	p. 1
0.2 Transformations	p. 3
0.3 Basic convexity	p. 6
0.4 The Hausdorff metric	p. 9
0.5 Measure and integration	p. 10
0.6 The support function	p. 16
0.7 Star sets and the radial function	p. 18
0.8 Polar duality	p. 20
0.9 Differentiability properties	p. 22
1 Parallel X-rays of planar convex bodies	p. 28
1.1 What is an X-ray?	p. 28
1.2 X-rays and Steiner symmetrals of planar convex bodies	p. 29
Open problems	p. 52
Notes	p. 52
1.1 Computerized tomography	p. 52
1.2 Parallel X-rays and Steiner symmetrals of convex bodies	p. 54
1.3 Exact reconstruction	p. 55
1.4 Well-posedness and stability	p. 56
1.5 Reconstruction of convex bodies from possibly noisy data	p. 57
1.6 Geometric probing	p. 58
1.7 Jakob Steiner (1796-1863)	p. 59
2 Parallel X-rays in n dimensions	p. 60
2.1 Parallel X-rays and k-symmetrals	p. 61
2.2 X-rays of convex bodies in E[superscript n]	p. 65
2.3 X-rays of bounded measurable sets	p. 69
Open problems	p. 86
Notes	p. 87
2.1 Parallel X-rays and k-symmetrals of convex bodies	p. 87
2.2 Switching components and discrete tomography	p. 88
2.3 Parallel X-rays and k-symmetrals of measurable sets	p. 91
2.4 Blaschke shaking	p. 92
2.5 Reconstruction of polygons and polyhedra from possibly noisy X-rays	p. 93
2.6 Ridge functions and the additivity conjecture	p. 94
2.7 X-rays of bounded density functions	p. 94
2.8 Johann Radon (1887-1956)	p. 95
3 Projections and projection functions	p. 97
3.1 Homothetic and similar projections	p. 98
3.2 The width function and central symmetral	p. 106
3.3 Projection functions and the Blaschke body	p. 110
Open problems	p. 125
Notes	p. 126
3.1 Homothetic and similar projections	p. 126
3.2 Bodies with congruent or affinely equivalent projections	p. 128
3.3 Sets of constant width and brightness	p. 129
3.4 Blaschke bodies and Blaschke sums	p. 130
3.5 Determination by one projection function	p. 131
3.6 Determination by more than one projection function	p. 132
3.7 Determination by directed projection functions, etc	p. 134
3.8 Reconstruction	p. 135
3.9 Mean projection bodies	p. 137
3.10 Projections of convex polytopes	p. 137
3.11 Critical projections	p. 138
3.12 Almost-spherical or almost-ellipsoidal projections, and related results	p. 138
3.13 Aleksander Danilovich Aleksandrov (1912-1999)	p. 139
4 Projection bodies and volume inequalities	p. 141
4.1 Projection bodies and related concepts	p. 142
4.2 Smaller bodies with larger projections	p. 154
4.3 Stability	p. 164
4.4 Reconstruction from brightness functions	p. 171
Open problems	p. 180
Notes	p. 180
4.1 Projection bodies and zonoids	p. 180
4.2 The Fourier transform approach I: The brightness function and projection bodies	p. 182
4.3 The Minkowski map and Minkowski linear combinations of projection bodies	p. 183
4.4 Generalized zonoids	p. 184
4.5 Bodies whose projections are zonoids	p. 185
4.6 Projection bodies of order i	p. 185
4.7 The L[superscript p]-Brunn-Minkowski theory and L[superscript p]-projection bodies	p. 186
4.8 Characterizations in terms of mixed volumes	p. 186
4.9 Results related to Aleksandrov's projection theorem	p. 187
4.10 Smaller bodies with larger projections	p. 187
4.11 Stability results	p. 189
4.12 Reconstruction from brightness functions	p. 190
4.13 Hermann Minkowski (1864-1909)	p. 192
5 Point X-rays	p. 194
5.1 Point X-rays and chordal symmetrals	p. 195
5.2 The X-ray of order i	p. 201
5.3 Point X-rays of planar convex bodies	p. 206
5.4 X-rays in the projective plane	p. 221
Open problems	p. 225
Notes	p. 226
5.1 Point X-rays and chordal symmetrals	p. 226
5.2 Point X-rays of planar convex bodies	p. 226
5.3 Reconstruction	p. 227
5.4 Well-posedness	p. 228
5.5 Point X-rays in higher dimensions	p. 228
5.6 Discrete point X-rays	p. 228
5.7 Point projections	p. 229
5.8 Wilhelm Suss (1895-1958) and the Japanese school	p. 229
6 Chord functions and equichordal problems	p. 232
6.1 i-chord functions and i-chordal symmetrals	p. 233
6.2 Chord functions of star sets	p. 237
6.3 Equichordal problems	p. 254
Open problems	p. 264
Notes	p. 264
6.1 Chord functions, i-chordal symmetrals, and ith radial sums	p. 264
6.2 Chord functions of star sets	p. 265
6.3 Equichordal problems	p. 265
6.4 Wilhelm Blaschke (1885-1962)	p. 267
7 Sections, section functions, and point X-rays	p. 269
7.1 Homothetic and similar sections	p. 270
7.2 Section functions and point X-rays	p. 276
7.3 Point X-rays of measurable sets	p. 286
Open problems	p. 288
Notes	p. 289
7.1 Homothetic and similar sections	p. 289
7.2 Bodies with congruent or affinely equivalent sections	p. 290
7.3 Sets of constant section	p. 290
7.4 Determination by section functions	p. 290
7.5 Determination by half-volumes	p. 291
7.6 Point X-rays of measurable sets	p. 293
7.7 Sections of convex polytopes	p. 294
7.8 Critical sections	p. 295
7.9 Almost-spherical or almost-ellipsoidal sections	p. 296
7.10 A characterization of star-shaped sets	p. 297
7.11 Sections by other sets of planes	p. 297
7.12 Integral geometry	p. 299
7.13 Mean section bodies	p. 300
7.14 Stereology	p. 301
7.15 Local stereology	p. 301
7.16 Paul Funk (1886-1969)	p. 303
8 Intersection bodies and volume inequalities	p. 304
8.1 Intersection bodies of star bodies	p. 305
8.2 Larger bodies with smaller sections	p. 324
8.3 Cross-section bodies	p. 334
Open problems	p. 336
Notes	p. 337
8.1 Intersection bodies	p. 337
8.2 The Fourier transform approach II: The section function and intersection bodies	p. 340
8.3 The map I	p. 340
8.4 Generalized intersection bodies	p. 340
8.5 Bodies whose central sections are intersection bodies	p. 341
8.6 Intersection bodies of order i	p. 341
8.7 k-intersection bodies and related notions	p. 341
8.8 Characterizations in terms of dual mixed volumes	p. 342
8.9 Larger bodies with smaller sections I: The Busemann-Petty problem	p. 343
8.10 Larger bodies with smaller sections II: Generalizations and variants of the Busemann-Petty problem	p. 345
8.11 Stability results	p. 346
8.12 Cross-section bodies	p. 346
8.13 Problems involving both projections and sections	p. 348
8.14 Herbert Busemann (1905-1994)	p. 348
9 Estimates from projection and section functions	p. 350
9.1 Centroid bodies	p. 351
9.2 Some affine isoperimetric inequalities	p. 355
9.3 Volume estimates from projection functions	p. 362
9.4 Volume estimates from section functions	p. 370
Open problems	p. 375
Notes	p. 376
9.1 Centroid bodies and polar projection bodies	p. 376
9.2 The floating body problem	p. 376
9.3 Affine surface area, the covariogram, and convolution and sectional bodies	p. 377
9.4 Affine isoperimetric inequalities	p. 379
9.5 The L[superscript p]-Brunn-Minkowski theory: centroid bodies, ellipsoids, and inequalities	p. 380
9.6 Volume estimates from projection functions	p. 382
9.7 Volume estimates from section functions	p. 384
9.8 The slicing problem	p. 385
9.9 Central limit theorems for convex bodies	p. 387
9.10 Estimates concerning both projections and sections	p. 388
9.11 Estimates for inradius and circumradius	p. 389
9.12 Hugo Hadwiger (1908-1981)	p. 389
Appendixes
A Mixed volumes and dual mixed volumes	p. 391
A.1 An example	p. 392
A.2 Area measures	p. 395
A.3 Mixed volumes and mixed area measures	p. 397
A.4 Reconstruction from surface area measures	p. 401
A.5 Quermassintegrals and intrinsic volumes	p. 403
A.6 Projection formulas	p. 406
A.7 Dual mixed volumes	p. 409
B Inequalities	p. 413
B.1 Inequalities involving means and sums	p. 413
B.2 The Brunn-Minkowski inequality	p. 415
B.3 The Aleksandrov-Fenchel inequality	p. 419
B.4 The dual Aleksandrov-Fenchel inequality	p. 420
B.5 Other inequalities	p. 422
C Integral transforms	p. 424
C.1 X-ray transforms	p. 424
C.2 The cosine and spherical Radon transforms	p. 427
Open problem	p. 436
References	p. 437
Notation	p. 471
Author index	p. 476
Subject index	p. 482

Available:*

On Order

Summary

Summary

Table of Contents