Convex Cones

This book provides the foundations for geometric applications of convex cones and presents selected examples from a wide range of topics, including polytope theory, stochastic geometry, and Brunn?Minkowski theory. Giving an introduction to convex cones, it describes their most important geometric functionals, such as conic intrinsic volumes and Grassmann angles, and develops general versions of the relevant formulas, namely the Steiner formula and kinematic formula.

In recent years questions related to convex cones have arisen in applied mathematics, involving, for example, properties of random cones and their non-trivial intersections. The prerequisites for this work, such as integral geometric formulas and results on conic intrinsic volumes, were previously scattered throughout the literature, but no coherent presentation was available. The present book closes this gap. It includes several pearls from the theory of convex cones, which should be better known.

?Convex cones: geometry and probability stands out in the present literature as probably the first and only textbook entirely dedicated to the development of the theory of convex cones ? . Given the self-contained nature of the presentation and the number of results and research directions that are touched upon ? this book seems bound to serve as a useful reference point for a number of research mathematicians in the years to come.? (Giorgos Chasapis, Mathematical Reviews, December, 2023)

?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.? (Florian Pausinger, zbMATH 1505.52001, 2023)