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| Preface to the second edition | |
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| Preface | |
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| Background material | |
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| Basic concepts and terminology | |
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| Transformations | |
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| Basic convexity | |
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| The Hausdorff metric | |
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| Measure and integration | |
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| The support function | |
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| Star sets and the radial function | |
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| Polar duality | |
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| Differentiability properties | |
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| Parallel X-rays of planar convex bodies | |
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| What is an X-ray? | |
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| X-rays and Steiner symmetrals of planar convex bodies | |
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| Open problems | |
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| Notes | |
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| Computerized tomography | |
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| Parallel X-rays and Steiner symmetrals of convex bodies | |
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| Exact reconstruction | |
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| Well-posedness and stability | |
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| Reconstruction of convex bodies from possibly noisy data | |
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| Geometric probing | |
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| Jakob Steiner (1796-1863) | |
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| Parallel X-rays in n dimensions | |
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| Parallel X-rays and k-symmetrals | |
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| X-rays of convex bodies in E[superscript n] | |
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| X-rays of bounded measurable sets | |
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| Open problems | |
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| Notes | |
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| Parallel X-rays and k-symmetrals of convex bodies | |
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| Switching components and discrete tomography | |
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| Parallel X-rays and k-symmetrals of measurable sets | |
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| Blaschke shaking | |
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| Reconstruction of polygons and polyhedra from possibly noisy X-rays | |
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| Ridge functions and the additivity conjecture | |
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| X-rays of bounded density functions | |
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| Johann Radon (1887-1956) | |
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| Projections and projection functions | |
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| Homothetic and similar projections | |
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| The width function and central symmetral | |
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| Projection functions and the Blaschke body | |
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| Open problems | |
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| Notes | |
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| Homothetic and similar projections | |
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| Bodies with congruent or affinely equivalent projections | |
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| Sets of constant width and brightness | |
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| Blaschke bodies and Blaschke sums | |
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| Determination by one projection function | |
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| Determination by more than one projection function | |
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| Determination by directed projection functions, etc | |
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| Reconstruction | |
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| Mean projection bodies | |
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| Projections of convex polytopes | |
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| Critical projections | |
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| Almost-spherical or almost-ellipsoidal projections, and related results | |
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| Aleksander Danilovich Aleksandrov (1912-1999) | |
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| Projection bodies and volume inequalities | |
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| Projection bodies and related concepts | |
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| Smaller bodies with larger projections | |
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| Stability | |
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| Reconstruction from brightness functions | |
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| Open problems | |
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| Notes | |
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| Projection bodies and zonoids | |
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| The Fourier transform approach I: The brightness function and projection bodies | |
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| The Minkowski map and Minkowski linear combinations of projection bodies | |
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| Generalized zonoids | |
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| Bodies whose projections are zonoids | |
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| Projection bodies of order i | |
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| The L[superscript p]-Brunn-Minkowski theory and L[superscript p]-projection bodies | |
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| Characterizations in terms of mixed volumes | |
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| Results related to Aleksandrov's projection theorem | |
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| Smaller bodies with larger projections | |
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| Stability results | |
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| Reconstruction from brightness functions | |
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| Hermann Minkowski (1864-1909) | |
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| Point X-rays | |
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| Point X-rays and chordal symmetrals | |
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| The X-ray of order i | |
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| Point X-rays of planar convex bodies | |
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| X-rays in the projective plane | |
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| Open problems | |
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| Notes | |
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| Point X-rays and chordal symmetrals | |
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| Point X-rays of planar convex bodies | |
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| Reconstruction | |
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| Well-posedness | |
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| Point X-rays in higher dimensions | |
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| Discrete point X-rays | |
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| Point projections | |
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| Wilhelm Suss (1895-1958) and the Japanese school | |
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| Chord functions and equichordal problems | |
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| i-chord functions and i-chordal symmetrals | |
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| Chord functions of star sets | |
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| Equichordal problems | |
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| Open problems | |
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| Notes | |
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| Chord functions, i-chordal symmetrals, and ith radial sums | |
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| Chord functions of star sets | |
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| Equichordal problems | |
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| Wilhelm Blaschke (1885-1962) | |
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| Sections, section functions, and point X-rays | |
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| Homothetic and similar sections | |
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| Section functions and point X-rays | |
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| Point X-rays of measurable sets | |
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| Open problems | |
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| Notes | |
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| Homothetic and similar sections | |
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| Bodies with congruent or affinely equivalent sections | |
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| Sets of constant section | |
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| Determination by section functions | |
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| Determination by half-volumes | |
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| Point X-rays of measurable sets | |
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| Sections of convex polytopes | |
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| Critical sections | |
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| Almost-spherical or almost-ellipsoidal sections | |
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| A characterization of star-shaped sets | |
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| Sections by other sets of planes | |
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| Integral geometry | |
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| Mean section bodies | |
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| Stereology | |
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| Local stereology | |
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| Paul Funk (1886-1969) | |
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| Intersection bodies and volume inequalities | |
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| Intersection bodies of star bodies | |
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| Larger bodies with smaller sections | |
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| Cross-section bodies | |
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| Open problems | |
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| Notes | |
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| Intersection bodies | |
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| The Fourier transform approach II: The section function and intersection bodies | |
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| The map I | |
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| Generalized intersection bodies | |
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| Bodies whose central sections are intersection bodies | |
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| Intersection bodies of order i | |
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| k-intersection bodies and related notions | |
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| Characterizations in terms of dual mixed volumes | |
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| Larger bodies with smaller sections I: The Busemann-Petty problem | |
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| Larger bodies with smaller sections II: Generalizations and variants of the Busemann-Petty problem | |
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| Stability results | |
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| Cross-section bodies | |
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| Problems involving both projections and sections | |
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| Herbert Busemann (1905-1994) | |
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| Estimates from projection and section functions | |
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| Centroid bodies | |
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| Some affine isoperimetric inequalities | |
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| Volume estimates from projection functions | |
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| Volume estimates from section functions | |
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| Open problems | |
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| Notes | |
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| Centroid bodies and polar projection bodies | |
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| The floating body problem | |
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| Affine surface area, the covariogram, and convolution and sectional bodies | |
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| Affine isoperimetric inequalities | |
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| The L[superscript p]-Brunn-Minkowski theory: centroid bodies, ellipsoids, and inequalities | |
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| Volume estimates from projection functions | |
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| Volume estimates from section functions | |
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| The slicing problem | |
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| Central limit theorems for convex bodies | |
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| Estimates concerning both projections and sections | |
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| Estimates for inradius and circumradius | |
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| Hugo Hadwiger (1908-1981) | |
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| Appendixes | |
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| Mixed volumes and dual mixed volumes | |
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| An example | |
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| Area measures | |
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| Mixed volumes and mixed area measures | |
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| Reconstruction from surface area measures | |
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| Quermassintegrals and intrinsic volumes | |
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| Projection formulas | |
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| Dual mixed volumes | |
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| Inequalities | |
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| Inequalities involving means and sums | |
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| The Brunn-Minkowski inequality | |
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| The Aleksandrov-Fenchel inequality | |
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| The dual Aleksandrov-Fenchel inequality | |
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| Other inequalities | |
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| Integral transforms | |
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| X-ray transforms | |
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| The cosine and spherical Radon transforms | |
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| Open problem | |
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| References | |
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| Notation | |
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| Author index | |
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| Subject index | |