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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.
Preface
Basic notions and facts
Notation and prerequisites
Incidence algebras
Convex cones
Polyhedra
Recession cones
Valuations
Identities for characteristic functions
Polarity as a valuation
A characterization of polarity
Angle functions
Invariant measures
Angles
Conic intrinsic volumes and Grassmann angles
Polyhedral Gauss–Bonnet theorems
A tube formula for compact general polyhedra
Relations to spherical geometry
Basic facts
The gnomonic map
Spherical and conic valuations
Inequalities in spherical space
Steiner and kinematic formulas
A general Steiner formula for polyhedral cones
The local Gaussian Steiner formula
The local spherical Steiner formula
Support measures of general convex cones
Kinematic formulas
Concentration of the conic intrinsic volumes
Inequalities and monotonicity properties
Observations about the conic support measures
Central hyperplane arrangements and induced cones
The Klivans–Swartz formula
Absorption probabilities via central arrangements
Random cones generated by central arrangements
Volume weighted Schl¨afli cones
Typical faces
Intersections of random cones
Miscellanea on random cones
Random projections
Gaussian images of cones
Wendel probabilities in high dimensions
Donoho–Tanner cones in high dimensions
Cover–Efron cones in high dimensions
Random cones in halfspaces
Convex hypersurfaces adapted to cones
Coconvex sets
Mixed volumes involving bounded coconvex sets
Wulff shapes in cones
A Minkowski-type existence theorem
A Brunn–Minkowski theorem for coconvex sets
Mixed volumes of general coconvex sets
Minkowski’s theorem for general coconvex sets
The cone-volume measure
Appendix: Open questions
References
Notation Index
Author Index
Subject Index