Convex Cones, 1st ed. 2022 Geometry and Probability Lecture Notes in Mathematics Series, Vol. 2319

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.

Basic notions and facts.- Angle functions.- Relations to spherical geometry.- Steiner and kinematic formulas.- Central hyperplane arrangements and induced cones.- Miscellanea on random cones.- Convex hypersurfaces adapted to cones.

Rolf Schneider is Professor Emeritus at the University of Freiburg. He obtained his PhD in 1967 (Frankfurt) and his Habilitation in 1969 (Bochum), after which he was Assistant Professor at the University of Frankfurt (1970) and then Full Professor at Technische Universität Berlin (1970) and the University of Freiburg (1974). He became Professor Emeritus in 2005. He is an Honorary Doctor of the University of Salzburg and a Fellow of the American Mathematical Society. With research interests primarily in convex geometry and stochastic geometry, he has over 200 publications, including the books Convex Bodies: The Brunn–Minkowski Theory and (with Wolfgang Weil) Stochastic and Integral Geometry.