Convex Cones

Geometry and Probability

verfasst von: Rolf Schneider

Verlag: Springer International Publishing

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

1. Basic notions and facts

Abstract

After fixing the notation and recalling some basic facts about incidence algebras, which will occasionally be needed, we use Section 1.3 to present introductory material about closed convex cones. Here we provide also some special lemmas, which will later be applied. Section 1.4 is devoted to polyhedra and deals with their normal cones and angle cones. In Section 1.5 we consider recession cones and show how they can be used in the description of unbounded polyhedra.

Rolf Schneider

2. Angle functions

Abstract

Whereas the considerations of the first chapter were essentially combinatorial in character, we begin now with measuring convex polytopes and polyhedral cones. In Section 2.1 we deal briefly with invariant measures, as needed later.

Rolf Schneider

3. Relations to spherical geometry

Abstract

Rolf Schneider

4. Steiner and kinematic formulas

Abstract

The conic intrinsic volumes and their spherical counterparts can alternatively be introduced by means of a so-called Steiner formula. Such a formula expresses the (Euclidean, Gaussian, or spherical) volume of a parallel set of a suitable given set at a given distance ε as a function of ε, exhibiting a special form from which functionals depending only on the given set can be extracted.

Rolf Schneider

5. Central hyperplane arrangements and induced cones

Abstract

The subsequent sections of this chapter deal with random cones generated by random central hyperplane arrangements. This topic was initiated a long time ago by Cover and Efron [50]. Their work is expanded considerably in Sections 5.3–5.5.

Rolf Schneider

6. Miscellanea on random cones

Abstract

A natural way to generate a random cone is to apply a random linear map to a fixed cone. If the distribution of the random linear mapping allows it, it may be possible to obtain explicit results for some expected geometric functionals of the random cone. The brief Section 6.1 deals with uniform random orthogonal projections of polyhedral cones (or general convex polyhedra). Section 6.2 treats images of general convex cones under linear maps defined by Gaussian matrices.

Rolf Schneider

7. Convex hypersurfaces adapted to cones

Abstract

In this chapter, the viewpoint is distinctly different. We still start with a pointed closed convex cone C with interior points. But our main interest will be in convex hypersurfaces, namely boundaries of closed convex sets, in this cone, whose behavior at infinity is determined by the cone.

Rolf Schneider

8. Appendix: Open questions

Abstract

We have occasionally mentioned open questions, and in this Appendix we want to repeat them and present them as a brief collection, for the reader’s convenience.

Rolf Schneider

Backmatter

Titel: Convex Cones
verfasst von: Rolf Schneider
Verlag: Springer International Publishing
Electronic ISBN: 978-3-031-15127-9
Print ISBN: 978-3-031-15126-2
DOI: https://doi.org/10.1007/978-3-031-15127-9