| Preface to the second edition | p. xv |
| Preface | p. xvii |
| Background material | p. 1 |
| Basic concepts and terminology | p. 1 |
| Transformations | p. 3 |
| Basic convexity | p. 6 |
| The Hausdorff metric | p. 9 |
| Measure and integration | p. 10 |
| The support function | p. 16 |
| Star sets and the radial function | p. 18 |
| Polar duality | p. 20 |
| Differentiability properties | p. 22 |
| Parallel X-rays of planar convex bodies | p. 28 |
| What is an X-ray? | p. 28 |
| X-rays and Steiner symmetrals of planar convex bodies | p. 29 |
| Open problems | p. 52 |
| Notes | p. 52 |
| Computerized tomography | p. 52 |
| Parallel X-rays and Steiner symmetrals of convex bodies | p. 54 |
| Exact reconstruction | p. 55 |
| Well-posedness and stability | p. 56 |
| Reconstruction of convex bodies from possibly noisy data | p. 57 |
| Geometric probing | p. 58 |
| Jakob Steiner (1796-1863) | p. 59 |
| Parallel X-rays in n dimensions | p. 60 |
| Parallel X-rays and k-symmetrals | p. 61 |
| X-rays of convex bodies in E[superscript n] | p. 65 |
| X-rays of bounded measurable sets | p. 69 |
| Open problems | p. 86 |
| Notes | p. 87 |
| Parallel X-rays and k-symmetrals of convex bodies | p. 87 |
| Switching components and discrete tomography | p. 88 |
| Parallel X-rays and k-symmetrals of measurable sets | p. 91 |
| Blaschke shaking | p. 92 |
| Reconstruction of polygons and polyhedra from possibly noisy X-rays | p. 93 |
| Ridge functions and the additivity conjecture | p. 94 |
| X-rays of bounded density functions | p. 94 |
| Johann Radon (1887-1956) | p. 95 |
| Projections and projection functions | p. 97 |
| Homothetic and similar projections | p. 98 |
| The width function and central symmetral | p. 106 |
| Projection functions and the Blaschke body | p. 110 |
| Open problems | p. 125 |
| Notes | p. 126 |
| Homothetic and similar projections | p. 126 |
| Bodies with congruent or affinely equivalent projections | p. 128 |
| Sets of constant width and brightness | p. 129 |
| Blaschke bodies and Blaschke sums | p. 130 |
| Determination by one projection function | p. 131 |
| Determination by more than one projection function | p. 132 |
| Determination by directed projection functions, etc | p. 134 |
| Reconstruction | p. 135 |
| Mean projection bodies | p. 137 |
| Projections of convex polytopes | p. 137 |
| Critical projections | p. 138 |
| Almost-spherical or almost-ellipsoidal projections, and related results | p. 138 |
| Aleksander Danilovich Aleksandrov (1912-1999) | p. 139 |
| Projection bodies and volume inequalities | p. 141 |
| Projection bodies and related concepts | p. 142 |
| Smaller bodies with larger projections | p. 154 |
| Stability | p. 164 |
| Reconstruction from brightness functions | p. 171 |
| Open problems | p. 180 |
| Notes | p. 180 |
| Projection bodies and zonoids | p. 180 |
| The Fourier transform approach I: The brightness function and projection bodies | p. 182 |
| The Minkowski map and Minkowski linear combinations of projection bodies | p. 183 |
| Generalized zonoids | p. 184 |
| Bodies whose projections are zonoids | p. 185 |
| Projection bodies of order i | p. 185 |
| The L[superscript p]-Brunn-Minkowski theory and L[superscript p]-projection bodies | p. 186 |
| Characterizations in terms of mixed volumes | p. 186 |
| Results related to Aleksandrov's projection theorem | p. 187 |
| Smaller bodies with larger projections | p. 187 |
| Stability results | p. 189 |
| Reconstruction from brightness functions | p. 190 |
| Hermann Minkowski (1864-1909) | p. 192 |
| Point X-rays | p. 194 |
| Point X-rays and chordal symmetrals | p. 195 |
| The X-ray of order i | p. 201 |
| Point X-rays of planar convex bodies | p. 206 |
| X-rays in the projective plane | p. 221 |
| Open problems | p. 225 |
| Notes | p. 226 |
| Point X-rays and chordal symmetrals | p. 226 |
| Point X-rays of planar convex bodies | p. 226 |
| Reconstruction | p. 227 |
| Well-posedness | p. 228 |
| Point X-rays in higher dimensions | p. 228 |
| Discrete point X-rays | p. 228 |
| Point projections | p. 229 |
| Wilhelm Suss (1895-1958) and the Japanese school | p. 229 |
| Chord functions and equichordal problems | p. 232 |
| i-chord functions and i-chordal symmetrals | p. 233 |
| Chord functions of star sets | p. 237 |
| Equichordal problems | p. 254 |
| Open problems | p. 264 |
| Notes | p. 264 |
| Chord functions, i-chordal symmetrals, and ith radial sums | p. 264 |
| Chord functions of star sets | p. 265 |
| Equichordal problems | p. 265 |
| Wilhelm Blaschke (1885-1962) | p. 267 |
| Sections, section functions, and point X-rays | p. 269 |
| Homothetic and similar sections | p. 270 |
| Section functions and point X-rays | p. 276 |
| Point X-rays of measurable sets | p. 286 |
| Open problems | p. 288 |
| Notes | p. 289 |
| Homothetic and similar sections | p. 289 |
| Bodies with congruent or affinely equivalent sections | p. 290 |
| Sets of constant section | p. 290 |
| Determination by section functions | p. 290 |
| Determination by half-volumes | p. 291 |
| Point X-rays of measurable sets | p. 293 |
| Sections of convex polytopes | p. 294 |
| Critical sections | p. 295 |
| Almost-spherical or almost-ellipsoidal sections | p. 296 |
| A characterization of star-shaped sets | p. 297 |
| Sections by other sets of planes | p. 297 |
| Integral geometry | p. 299 |
| Mean section bodies | p. 300 |
| Stereology | p. 301 |
| Local stereology | p. 301 |
| Paul Funk (1886-1969) | p. 303 |
| Intersection bodies and volume inequalities | p. 304 |
| Intersection bodies of star bodies | p. 305 |
| Larger bodies with smaller sections | p. 324 |
| Cross-section bodies | p. 334 |
| Open problems | p. 336 |
| Notes | p. 337 |
| Intersection bodies | p. 337 |
| The Fourier transform approach II: The section function and intersection bodies | p. 340 |
| The map I | p. 340 |
| Generalized intersection bodies | p. 340 |
| Bodies whose central sections are intersection bodies | p. 341 |
| Intersection bodies of order i | p. 341 |
| k-intersection bodies and related notions | p. 341 |
| Characterizations in terms of dual mixed volumes | p. 342 |
| Larger bodies with smaller sections I: The Busemann-Petty problem | p. 343 |
| Larger bodies with smaller sections II: Generalizations and variants of the Busemann-Petty problem | p. 345 |
| Stability results | p. 346 |
| Cross-section bodies | p. 346 |
| Problems involving both projections and sections | p. 348 |
| Herbert Busemann (1905-1994) | p. 348 |
| Estimates from projection and section functions | p. 350 |
| Centroid bodies | p. 351 |
| Some affine isoperimetric inequalities | p. 355 |
| Volume estimates from projection functions | p. 362 |
| Volume estimates from section functions | p. 370 |
| Open problems | p. 375 |
| Notes | p. 376 |
| Centroid bodies and polar projection bodies | p. 376 |
| The floating body problem | p. 376 |
| Affine surface area, the covariogram, and convolution and sectional bodies | p. 377 |
| Affine isoperimetric inequalities | p. 379 |
| The L[superscript p]-Brunn-Minkowski theory: centroid bodies, ellipsoids, and inequalities | p. 380 |
| Volume estimates from projection functions | p. 382 |
| Volume estimates from section functions | p. 384 |
| The slicing problem | p. 385 |
| Central limit theorems for convex bodies | p. 387 |
| Estimates concerning both projections and sections | p. 388 |
| Estimates for inradius and circumradius | p. 389 |
| Hugo Hadwiger (1908-1981) | p. 389 |
| Appendixes | |
| Mixed volumes and dual mixed volumes | p. 391 |
| An example | p. 392 |
| Area measures | p. 395 |
| Mixed volumes and mixed area measures | p. 397 |
| Reconstruction from surface area measures | p. 401 |
| Quermassintegrals and intrinsic volumes | p. 403 |
| Projection formulas | p. 406 |
| Dual mixed volumes | p. 409 |
| Inequalities | p. 413 |
| Inequalities involving means and sums | p. 413 |
| The Brunn-Minkowski inequality | p. 415 |
| The Aleksandrov-Fenchel inequality | p. 419 |
| The dual Aleksandrov-Fenchel inequality | p. 420 |
| Other inequalities | p. 422 |
| Integral transforms | p. 424 |
| X-ray transforms | p. 424 |
| 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 |
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