Convex cones spanned by regular polytopes
-
Zakhar Kabluchko
and
Hauke Seidel
Abstract
We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic geometry can be expressed through these conic intrinsic volumes. A list of such quantities includes internal and external solid angles of regular simplices and crosspolytopes, the probability that a (symmetric) Gaussian random polytope or the Gaussian zonotope contains a given point, the expected number of faces of the intersection of a regular polytope with a random linear subspace passing through its centre, and the expected number of faces of the projection of a regular polytope onto a random linear subspace.
Acknowledgements
We thank Jonathan Thalmann for pointing out a mistake in a former version of Proposition 2.3 and the unknown referee for a careful reading of the manuscript.
Communicated by: M. Henk
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Articles in the same Issue
- Frontmatter
- Semistable Higgs bundles on elliptic surfaces
- On the rigidity of harmonic-Ricci solitons
- On Ricci negative derivations
- Approximating coarse Ricci curvature on submanifolds of Euclidean space
- Convex cones spanned by regular polytopes
- On the geography of line arrangements
- Towards resolving Keller’s cube tiling conjecture in dimension seven
Articles in the same Issue
- Frontmatter
- Semistable Higgs bundles on elliptic surfaces
- On the rigidity of harmonic-Ricci solitons
- On Ricci negative derivations
- Approximating coarse Ricci curvature on submanifolds of Euclidean space
- Convex cones spanned by regular polytopes
- On the geography of line arrangements
- Towards resolving Keller’s cube tiling conjecture in dimension seven
