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.
Author(s): Rolf Schneider
Series: Lecture Notes in Mathematics, 2319
Publisher: Springer
Year: 2022
Language: English
Pages: 351
City: Cham
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
