Convex cones. Geometry and probability (Q2170866)

This book develops the necessary machinery for geometric applications of convex cones. The concise style and wealth of references and ideas make this book without doubt a valuable source of information. Each chapter of the book starts with an overview that outlines the main ideas and results. Moreover, each section finishes with a list of notes containing further references, open problems and extensions of presented results. A central topic of the book are different geometric functionals of convex cones such as conic intrinsic volumes and Grassmann angles. Conic intrinsic volumes are the conic counterparts to the intrinsic volumes of convex bodies and their theory is introduced in Chapter 2. Chapter 3 investigates the relation between the geometry of subsets of the sphere and the geometry of cones, which turns out to be equivalent. However, depending on the concrete application one viewpoint may be preferred over the other. Importantly, general versions of relevant formulas such as the Steiner formula or the kinematic formula are developed in a systematic manner in Chapter 4. Furthermore, Chapter 5 looks into arrangements of finitely many hyperplanes through the origin, which naturally generate tessellations of the space into polyhedral convex cones. Here, a formula for the sum of the \(k\)-th conic intrinsic volumes of the \(j\)-faces of such tessellations is derived. Moreover, different models of random cones are studied based on random hyperplane arrangements with certain distributions subject to mild assumptions. The study of random cones is continued in Chapter 6 and pursued into various different directions. The final chapter of the book studies so called C-coconvex sets for which parts of a Brunn-Minkowski theory are developed. Finally, the book ends with a collection of all open questions that popped up at different places in the book.