# Intersection probabilities and kinematic formulas for polyhedral cones

## TL;DR

This paper proves a conic kinematic formula for polyhedral convex cones without using characterization theorems and calculates intersection probabilities for random cones from isotropic hyperplane arrangements.

## Contribution

It provides a new proof of the conic kinematic formula and derives intersection probabilities for specific classes of random cones, expanding understanding in convex geometry.

## Key findings

- Proof of conic kinematic formula avoiding characterization theorems
- Probabilities for non-trivial intersections of random cones
- Results applicable to isotropic hyperplane arrangements

## Abstract

For polyhedral convex cones in ${\mathbb R}^d$, we give a proof for the conic kinematic formula for conic curvature measures, which avoids the use of characterization theorems. For the random cones defined as typical cones of an isotropic random central hyperplane arrangement, we find probabilities for non-trivial intersection, either with a fixed cone, or for two independent random cones of this type.