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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 | |