Convex cones : geometry and probability / / Rolf Schneider

• Record Nr.: UNISA996490271403316
• Autore: Schneider Rolf <1940->
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 3-031-15127-5
• Descrizione fisica: 1 online resource (352 pages)
• Collana: Lecture Notes in Mathematics ; ; v.2319
• Disciplina: 929.605
• Soggetti: Convex bodies; Cossos convexos; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Preface -- Contents -- 1 Basic notions and facts -- 1.1 Notation and prerequisites -- 1.2 Incidence algebras -- 1.3 Convex cones -- 1.4 Polyhedra -- 1.5 Recession cones -- 1.6 Valuations -- 1.7 Identities for characteristic functions -- 1.8 Polarity as a valuation -- 1.9 A characterization of polarity -- 2 Angle functions -- 2.1 Invariant measures -- 2.2 Angles -- 2.3 Conic intrinsic volumes and Grassmann angles -- 2.4 Polyhedral Gauss-Bonnet theorems -- 2.5 A tube formula for compact general polyhedra -- 3 Relations to spherical geometry -- 3.1 Basic facts -- 3.2 The gnomonic map -- 3.3 Spherical and conic valuations -- 3.4 Inequalities in spherical space -- 4 Steiner and kinematic formulas -- 4.1 A general Steiner formula for polyhedral cones -- 4.1.1 The local Gaussian Steiner formula -- 4.1.2 The local spherical Steiner formula -- 4.2 Support measures of general convex cones -- 4.3 Kinematic formulas -- 4.4 Concentration of the conic intrinsic volumes -- 4.5 Inequalities and monotonicity properties -- 4.6 Observations about the conic support measures -- 5 Central hyperplane arrangements and induced cones -- 5.1 The Klivans-Swartz formula -- 5.2 Absorption probabilities via central arrangements -- 5.3 Random cones generated by central arrangements -- 5.4 Volume weighted Schl¨afli cones -- 5.5 Typical faces -- 5.6 Intersections of random cones -- 6 Miscellanea on random cones -- 6.1 Random ; projections -- 6.2 Gaussian images of cones -- 6.3 Wendel probabilities in high dimensions -- 6.4 Donoho-Tanner cones in high dimensions -- 6.5 Cover-Efron cones in high dimensions -- 6.6 Random cones in halfspaces -- 7 Convex hypersurfaces adapted to cones -- 7.1 Coconvex sets -- 7.2 Mixed volumes involving bounded coconvex sets -- 7.3 Wulff shapes in cones -- 7.4 A Minkowski-type existence theorem -- 7.5 A Brunn-Minkowski theorem for coconvex sets.; 7.6 Mixed volumes of general coconvex sets -- 7.7 Minkowski's theorem for general coconvex sets -- 7.8 The cone-volume measure -- 8 Appendix: Open questions -- References -- Notation Index -- Author Index -- Subject Index.