Convex bodies are at once simple and amazingly rich in structure. While the classical results go back many decades, during the past ten years the integral geometry of convex bodies has undergone a dramatic revitalization, brought about by the introduction of methods, results and, most importantly, new viewpoints, from probability theory, harmonic analysis and the geometry of finite-dimensional normed spaces. This collection arises from an MSRI program held in the Spring of 1996, involving researchers in classical convex geometry, geometric functional analysis, computational geometry, and related areas of harmonic analysis. It is representative of the best research in a very active field that brings together ideas from several major strands in mathematics.

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1. Integrals of smooth and analytic functions over Minkowski's sums of convex sets S. Alesker; 2. On the Gromov–Milman theorem on concentration phenomenon on the uniformly convex sphere S. Alesker; 3. Geometric inequalities in option pricing Christer Borell; 4. Random points in isotropic convex sets Jean Bourgain; 5. Threshold intervals under group symmetries Jean Bourgain and G. Kalai; 6. On a generalization of the Busemann–Petty problem Jean Bourgain and Gaoyong Zhang; 7. Isotropic constants of Schatten class spaces Sean Dar; 8. On the stability of the volume radius E. D. Gluskin; 9. Polytope approximations of the unit ball of Lpn W. T. Gowers; 10. A remark about the scalar-plus-compact problem W. T. Gowers; 11. Another low-technology estimate in convex geometry Greg Kuperberg; 12. On the equivalence between geometric and arithmetic means for log-concave measures Rafal Latala; 13. On the constant in the Reverse Brunn–Minkowski inequality for p-convex balls A. E. Litvak; 14. An extension of Krivine's theorem to quasi-normed spaces A. E. Litvak; 15. A note on Gowersí dichotomy theorem Bernard Maurey; 16. An isomorphic version of Dvoretzky's theorem II Vitali Milman and Gideon Schechtman; 17. Asymptotic versions of operators and operator ideals V. Milman and R. Wagner; 18. Metric entropy of the Grassman manifold Alain Pajor; 19. Curvature of nonlocal Markov generators Michael Schmuckenschlager; 20. An external property of the regular simplex Michael Schmuckenschlager; 21. Floating body, illumination body, and polytopal approximation Carsten Schutt; 22. A note on the M*-limiting convolution body Antonis Tsolomitis.