Rolf Schneider

Bio: Rolf Schneider is an academic researcher from University of Freiburg. The author has contributed to research in topics: Convex body & Polytope. The author has an hindex of 38, co-authored 212 publications receiving 8801 citations. Previous affiliations of Rolf Schneider include Ruhr University Bochum & Goethe University Frankfurt.

Convex bodies : the Brunn-Minkowski theory

Abstract: 1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

Stochastic and Integral Geometry

Abstract: Foundations of Stochastic Geometry.- Prolog.- Random Closed Sets.- Point Processes.- Geometric Models.- Integral Geometry.- Averaging with Invariant Measures.- Extended Concepts of Integral Geometry.- Integral Geometric Transformations.- Selected Topics from Stochastic Geometry.- Some Geometric Probability Problems.- Mean Values for Random Sets.- Random Mosaics.- Non-stationary Models.- Facts from General Topology.- Invariant Measures.- Facts from Convex Geometry.

Valuations on convex bodies

Abstract: The investigation of functions on convex bodies which are valuations, or additive in Hadwiger’s sense, has always been of interest in particular parts of geometric convexity, and it has seen some progress in recent years. The occurrence of valuations in the theory of convex bodies can be traced back to the notion of volume in two essentially different ways. Firstly, the volume of convex bodies, being the restriction of a measure, is itself a valuation. This valuation property carries over to the functions which are deduced from volume in the Brunn-Minkowski theory, namely to mixed volumes, quermassintegrals, surface area functions, and others. Hadwiger’s celebrated characterizations of the quermassintegrals by the valuation and other properties were the culmination of a series of papers on valuations and at the same time the starting point for various subsequent investigations of functionals with similar properties.

Zonoids and Related Topics

Abstract: A zonoid is a convex body in euclidean space which can be approximated, in sense of the Hausdorff metric, by finite vector sums of line segments Several equivalent definitions are known, and zonoids appear in some surprisingly different contexts The purpose of the present survey article is twofold After giving some basic definitions and properties of zonotopes and zonoids, we describe the various ways in which zonoids enter the discussion of a number of seemingly unrelated topics The stress is here on interrelations between the geometry of convex bodies and other fields The second part treats those results and problems concerning zonoids and their generalizations which are of interest within the theory of convex bodies

Random projections of regular simplices

Abstract: Precise asymptotic formulae are obtained for the expected number ofk-faces of the orthogonal projection of a regularn-simplex inn-space onto a randomly chosen isotropic subspace of fixed dimension or codimension, as the dimensionn tends to infinity.

Optimal Transport: Old and New

Abstract: Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Convex bodies : the Brunn-Minkowski theory

Abstract: 1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

Random Fields and Geometry

Abstract: * Recasts topics in random fields by following a completely new way of handling both geometry and probability * Significant exposition of the work of others in the field * Presentation is clear and pedagogical * Excellent reference work as well as excellent work for self study This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined. The three parts to the monograph are quite distinct. Part I presents a user-friendly yet comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities. Part II gives a quick review of geometry, both integral and Riemannian, to provide the reader with the material needed for Part III, and to give some new results and new proofs of known results along the way. Topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory, and tube formulae are covered. In fact, this is the only concise, self-contained treatment of all of the above topics, which are necessary for the study of random fields. The new approach in Part III is devoted to the geometry of excursion sets of random fields and the related Euler characteristic approach to extremal probabilities. "Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory. These applications, to appear in a forthcoming volume, will cover areas as widespread as brain imaging, physical oceanography, and astrophysics.