CONTENTS -- Preface -- Foreword -- Volume inequalities for sets associated with convex bodies -- 1. Introduction and preliminaries -- 2. Main volume inequalities -- 3. The Rogers and Shephard method -- 4. Applications -- References -- Integral geometry and Alesker's theory of valuations -- 1. Introduction -- 2. Some naive algebra -- 3. Consequences of the Irreducibility Theorem -- 4. Alesker multiplication -- 5. Applications to the integral geometry of the orthogonal and unitary groups -- References -- Area and perimeter bisectors of planar convex sets -- 1. Introduction -- 2. Notation and some early results -- 3. Auerbach's results -- 4. Some integral geometry results -- References -- Radon Inversion: from lines to Grassmannians -- 1. The Funk Transform on the Sphere -- 2. Double Fibrations and Induced Transforms -- 3. Helgason's Inversion Formula -- 4. Radon Inversion for Pairs of Grassmannians -- 5. Inversion by Generalized Fractional Integrals -- References -- Valuations in the affine geometry of convex bodies -- 1. Introduction -- 2. Real valued valuations on polytopes -- 3. Real valued valuations on convex bodies -- 4. Tensor valued valuations -- 5. Convex body valued valuations -- 6. Star body valued valuations -- References -- Crofton measures in projective Finsler spaces -- 1. The Integral-geometric Approach to Hilbert's Fourth Problem -- 2. The Zonoid Equation -- 3. Projective Finsler Spaces -- 4. Notions of Area -- 5. Nonexistence and Existence of (Positive) Crofton Measures -- 6. More on Generalized Zonoids -- 7. Crofton Formulas in Smooth Projective Finsler Spaces -- 8. Crofton Measures in General Projective Finsler Spaces -- References -- Random methods in approximation of convex bodies -- References.

Some generalized maximum principles and their applications to Chern type problems -- 1. Introduction -- 2. Preliminaries -- 3. A Liouville Type Theorem for n>2 -- 4. A Liouville Type Theorem for 1 < n < 2 -- 5. Applications of Liouville type Theorem 4.2 for the case 1 < n < 2 -- 6. A Liouville Type Theorem for n = 1 -- 7. Further Remarks -- 8. Weak Generalized Maximum Principle -- References -- Floating bodies and illumination bodies -- 1. Introduction and Notations -- 2. Affine surface area -- References -- Applications of information theory to convex geometry -- 1. Introduction -- 2. Connections between geometry analysis and information theory -- 3. A little information theory -- 4. A little convex geometry -- 5. Matrices and ellipsoids naturally associated with a convex body -- 6. Geometric inequalities -- 7. Moment-entropy inequalities -- 8. Affine Sobolev inequalities -- References -- Containment measures in integral geometry -- 1. Introduction -- 2. Containment measures of domains in homogeneous spaces -- 3. Integral formulas for containment measures -- 4. Chord power integrals and containment measures -- 5. Sufficient conditions for containment of convex bodies and isoperimetric inequalities -- 6. Sufficient conditions for containment of smooth convex bodies -- 7. Geometric probabilities and containment measures -- 8. Isoperimetric problems of containment measures -- 9. Sufficient conditions for containment of domains in space of constant curvature -- References -- On the flag curvature and S-curvature in Finsler geometry -- 1. Notations and Definitions -- 2. Flag Curvature and S-Curvature -- 3. Funk Metric and Its Curvature Properties -- References -- Double chord-power integrals of a convex body and their applications -- 1. Introduction -- 2. Double chord-power integrals of a convex body.

3. Applications to a problem of geometric probability -- 4. Problems -- References -- Lp dual Brunn-Minkowski type inequalities -- 1 Introduction -- 2 Notation and preliminary -- 3 Main results -- References -- On the relations of a convex set and its profile -- 1. Introduction -- 2. Main results -- 3. The proof of the main results -- 4. Conclusion -- References -- Convex bodies with symmetric X-rays in two directions -- 1. Basic concepts and notation -- 2. Lemmas -- 3. Main results -- References -- The kinematic measure of a random line segment of fixed length within a trapezoid -- 1. Preliminaries and Notations -- 2. Main Results -- 3. Application of mQ(l) to geometric probability -- References.