Convex cones spanned by regular polytopes

TL;DR

This paper investigates three families of polyhedral cones generated by regular polytopes, computing their geometric properties and linking these to stochastic geometry quantities such as Gaussian polytope probabilities and face counts.

Contribution

It introduces explicit calculations of solid angles and conic intrinsic volumes for cones spanned by regular simplices, cubes, and crosspolytopes, connecting these to stochastic geometric measures.

Findings

Computed solid angles and conic intrinsic volumes for the cones.

Expressed stochastic geometry quantities in terms of these intrinsic volumes.

Linked geometric properties to probabilities involving Gaussian random polytopes.

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.