Titlebook: Convex Cones; Geometry and Probabi Rolf Schneider Book 2022 The Editor(s) (if applicable) and The Author(s), under exclusive license to Spr

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https://doi.org/10.33283/978-3-86298-640-8Whereas 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.

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Angle functions,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.

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Relations to spherical geometry,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.

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Central hyperplane arrangements and induced cones,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.

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Convex hypersurfaces adapted to cones,In this chapter, the viewpoint is distinctly different. We still start with a pointed closed convex cone . 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.

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Appendix: Open questions,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.

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https://doi.org/10.1007/978-3-031-15127-9valuation; conic support measure; Grassmann angle; Master Steiner formula; central hyperplane tessellati